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1 第 19 回 整数論サマースクール報告集 保型形式のリフティング 2011 年 9 月 5 日 ~9 月 9 日於静岡県田方群富士箱根ランド スコーレプラザホテル

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3 L- Langlands 70 (A) ( : ) (B) ( : : ) 19 i

4 第 19 回 (2011 年度 ) 整数論サマースクール 保型形式のリフティング プログラム 日時 2011 年 9 月 5 日 ( 月 ) から 9 月 9 日 ( 金 ) まで 場所富士箱根ランド スコーレプラザホテル 9 月 5 日 ( 月 ) 10:00-12:00 GL2 の保型形式と表現論 ( 軍司圭一 ) 14:00-15:30 志村対応その 1 ( 坂田裕 ) 15:45-17:15 志村対応その 2 ( 坂田裕 ) 17:30-18:30 accidental isogeny について ( 成田宏秋 ) 20:00-21:00 Siegel 保型形式と Hecke 作用素 ( 軍司圭一 ) 9 月 6 日 ( 火 ) 9:30-10:30 Weil 表現 ( 松本久義 ) 10:45-12:15 Howe 対応 ( 松本久義 ) 14:00-15:00 Jacobi 形式 ( 高瀬幸一 ) 15:15-16:45 Saito-Kurokawa リフト 1 ( 高瀬幸一 ) 17:00-18:30 Saito-Kurokawa リフト 2 ( 伊吹山知義 ) 20:00-21:30 院生 若手の時間 9 月 7 日 ( 水 ) 9:00-10:00 theta 関数の変換公式 ( 宮崎直 ) 10:15-12:15 Oda リフト ( 菅野孝史 ) 14:00-16:00 Borcherds リフト ( 青木宏樹 ) 16:15-17:00 Eisenstein 級数の Fourier 係数 ( 軍司圭一 ) 17:15-18:45 Ikeda リフト ( 河村尚明 ) 9 月 8 日 ( 木 ) 14:00-15:30 Langlands 関手性 ( 吉田敬之 ) 16:00-18:00 Seesaw dual pair と内積公式 ( 山名俊介 ) 9 月 9 日 ( 金 ) 9:00-10:30 theta 対応に現れる cohomological 表現 ( 早田孝博 ) 10:45-12:15 二つの楕円保型形式からの重さ半整数 Siegel 保型形式へのリフト ( 林田秀一 ) ii

5 北海道大学河村尚明室蘭工業大学桂田英典東北大学太田和惟小林真一長瀬聡宏廣瀬康隆宮城教育大学高瀬幸一山形大学早田孝博東京大学織田孝幸甲斐亘鍛治匠一鈴木航介平野雄一松本久義宮崎弘安芳木武仁東京工業大学内藤聡東京理科大学青木宏樹加塩朋和慶応義塾大学大槻玲小野雅隆萩原啓早稲田大学岡本亮彦兵藤史武広中由美子森澤貴之柳内武志早稲田大学高等学院坂田裕工学院大学斎藤正顕長谷川武博立教大学佐藤文広宮崎直成蹊大学石井卓若林功明治大学対馬龍司横浜国立大学原下秀士千葉工業大学軍司圭一杉山和成金沢大学菅野孝史名古屋大学伊東杏希子小倉一輝京都大学池田保石塚裕大大下達也岡田健佐々木健太竹森翔千田雅隆林芳樹安田正大吉田敬之京都産業大学槇山賢治山上敦士大阪大学伊吹山知義落合理北山秀隆喜友名朝也源嶋孝太杉山真吾林田秀一原隆兵庫慶則前田恵近畿大学菊田俊幸長岡昇勇大阪市立大学森本和輝山名俊介広島大学飯島優香川大学内藤浩忠徳島大学水野義紀九州大学池松泰彦小笠原健高田芽味三柴善範熊本大学成田宏秋鹿児島大学山内卓也以上 76 名敬称略 所属は参加申請時のとおり参加者リスト iii

6 1. GL 2 1 ( ) 2. Shimura 17 ( ) 3. Accidental 37 ( ) 4. Siegel Hecke 63 ( ) 5. Weil Howe duality 76 ( ) 6. Siegel Jacobi 92 ( ) 7. Saito Kurokawa lifting for level N 163 ( ) ( ) 9. Oda Lift 195 ( ) 10. Borcherds product 212 ( ) iv

7 11. Siegel Eisenstein Fourier 242 ( ) 12. Functoriality Principle 254 ( ) 13. Seesaw dual pair 270 ( ) 14. Theta Cohomological 290 ( ) 15. Siegel 302 ( ) 16. On v-adic periods of t-motives 322 ( ) v

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9 vi

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11 1 GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2, R) H f, ( ) f k g(z) = j(g, z) k a b f(g(z)), g =, j(g, z) = cz + d, g(z) = az + b c d cz + d j(g 1 g 2, z) = j(g 1, g 2 (z))j(g 2, z) (f, g) f k g G H 1

12 2 2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f cusp M k (Γ ), cusp S k (Γ ) 2.1 f M k (Γ ) cusp M y k/2 f(z) < M Γ Fourier Hecke f S k (Γ ) a(n) = O(n k/2 ) Eisenstein theta Fourier Γ Γ 0 (N) = {( a c ) } b SL(2, Z) d c 0 mod N ψ N Dirichlet ( ) ψ Γ 0 (N) ψ (( a c )) b = ψ(d) d N M k (Γ 0 (N), ψ) = {f M k (Γ (N)) f k γ = ψ(γ)f, γ Γ 0 (N)} Fourier Dirichlet L 2

13 3 2.2 (L 1) f S k (Γ 0 (N), ψ) s C Fourier f = n=1 a(n)e2πinz L(f, s) = a(n)n s n=1 f L Re(s) L Mellin 0 f(iy)y s 1 dy = = 0 a(n)e 2πny y s 1 dy n=1 a(n)(2πn) 1 s e 2πny (2πny) s 1 dy (2.1) n=1 = (2π) s Γ (s)l(s, f) 2.2 (1) L(s, f) s (2) Λ(s, f) = N s/2 (2π) s Γ (s)l(s, f) Λ(s, f) = i k Λ(k s, f) f(z) = ( Nz) k f( (Nz) 1 ) (3) Λ(s, f) C > 0 V (C) = {s C Re(s) < C} Hecke Fourier Hecke Hecke ( ) T p 1 0 Γ 0 (N) Γ 0 (N) Γ 0 (N) Fourier 0 p 2.3 f(z) = n=1 a(n)e2πinz S k (Γ 0 (N), ψ), T p f(z) i) (p, N) = 1 T p f(z) = n=1 ( ( )) n a(np) + ψ(p)p k 1 a e 2πinz, p ( p n a ( ) n = 0) p 3

14 4 ii) p N T p f(z) = a(np)e 2πinz. n=1 T p f S k (Γ 0 (N), ψ) Atkin-Lehner [AL] ii) U p - T p - (p, N) = 1 p T p Petersson S k (Γ 0 (N), ψ) T p (p N) Atkin-Lehner new form Sk new (Γ 0 (N), ψ) p N p T p 2.4 (L 2) f T p T p λ p L(s, f) = a(1) p (1 λ p p s + ψ(p)p k 1 2s ) 1 a(1) f Fourier e 2πiz Re(s) f S k (Γ 0 (N), ψ) T p L 2.4 Euler f L 2.2 f 2.4 Euler 2.4 L SL 2 Dirichlet Fourier Siegel 2.2 Dirichlet Koecher-Maass ( 1 ) f, g S k (Γ 0 (N), ψ) T p f g 3 SL 2 (R) 3.1 Γ = Γ 0 (N) f M k (Γ ) G = SL 2 (R) K = SO(2) G f H G/K H, g g(i) f G 4

15 5 3.1 f S k (Γ ) G φ f φ f (g) = j(g, i) k f(g(i)) φ f C (G) 3.1 φ f (1) φ f Γ - γ Γ φ f (γg) = φ f (g) ( ) cos θ sin θ (2) K r θ = φ f (gr θ ) = e ikθ φ f (g) sin θ cos θ (3) φ f C M φ f (g) < C g M ( g = Tr( t gg) ) ( ) (4) sl(2, C) X = 1 1 i X φ f = 0 (X φ f 2 i 1 ) f cusp (5) δ SL(2, Z) Γ δ = δ 1 Γ δ δ SL(2, Z) g G φ f (δng) dn = 0. N Γ δ \N {( ) } 1 a N = G a R 0 1 (4) 2.1 (1) f X Cauchy-Riemann (1) f, (2) K i, (3) 2.1 (3) f cusp ) (1), (2) (3) g G z = x + yi H ( x 1/2, z > 1) ( ) y 1/2 xy 1/2 g = δg z r θ, δ SL(2, Z), g z = 0 y 1/2 φ f (g) = y k/2 e ikθ f k δ(z) y < C g M f k δ Fourier n 0 (4) (5) N Γ δ = {( 1 hm 0 1 ) m Z } 5

16 6 h z = g(i) H ( ) ) 1 1 hn φ f (δng) dn = φ f (δ g dn N Γ δ \N ( ( ) ) ( ( ) k 1 1 hn 1 hn = f δ g(i) j δ g, i) dn = f(δ(z + hn))j(δ, z + hn) k j(g, i) k dn 1 = j(g, i) k (f k δ)(z + hn) dn 0 f k δ f cusp (1) φ f C (Γ \G) G g G ψ C (Γ \G) gψ(h) = ψ(hg) gψ C (Γ \G) (2) φ f K Cφ f 1 K (4) G 3.2 G Lie G Hilbert G G- (g, K)- Lie G K K- (Peter-Weyl ) G- (π, H) K- H = ˆ m(δ)h δ, δ K m(δ) < K- (admissible) K = SO(2) 1 m Z K r θ e imθ SO(2) Z 6

17 7 G Lie g 0 G g 0 G Hilbert (π, H) g 0 = Lie(G) H v H X g 0 d dt π(exp(tx))v π(exp(tx))v v = lim (3.1) t=0 t 0 t v H H g 0 H Lie g := g 0 R C G (π, H) G- g- K- (K G ) H K- H K = {v H dim(π(k)v) < } H K H = ˆ δ m(δ)h δ H K = δ m(δ)h δ H K H H K H dense g K H K (g, K)- H G- H K (g, K)- (g, K)- [Ma] f M k (Γ 0 (N), ψ) X φ f = 0 X ( ) g (3.1) z = x + yi H y 1/2 xy 1/2 g z = 0 y 1/2 g z (i) = z g G g = g z r θ φ f K r θ X r 1 θ X = 1 2 ( ) + i 2 ( = e 2iθ X g = g z ) ( ) + i A, B, C A φ f (g) = 1 (( ) ( )) d y 1/2 xy 1/2 e t 0 2 dt φ f t=0 0 y 1/2 0 e t = 1 d ( 2 dt e kt y k/2 f(x + iye 2t ) ) t=0 = k 2 yk/2 f(z) + y k/2+1 y f(z) 7

18 8 B φ f (g) = i (( ) ( )) d y 1/2 xy 1/2 1 t 2 dt φ f t=0 0 y 1/2 0 1 = i d 2 dt y k/2 f(x ty + iy) t=0 = iyk/2+1 2 x f(z) C φ f (g) = i (( ) ( )) d y 1/2 xy 1/ dt φ f t=0 0 y 1/2 t 1 = i ( d 2 dt ( ti + 1) k y k/2 f x ty ) t=0 t i y t = k 2 yk/2 f(z) iyk/2+1 2 x f(z) f ( X φ f (g) = iy k/2+1 x + i ) f(z) = 0 y 3.3 f S k (Γ 0 (N)) 2.1 φ f L 2 (Γ \G) G Hilbert L 2 (Γ \G) K- L 2 (Γ \G) K (g, K)- φ f (g, K)-? (g, K)- ( ) 0 1 H = i, X + = ( ) ( 1 i, X = 1 i i i 1 ) g = sl(2, C) SL 2 -triple [X +, X ] = H [H, X + ] = 2X + [H, X ] = 2X exp(iθh) = r θ K (g, K)- V v m r θ v = e imθ v Hv = mv X + v, X v m + 2, m 2 k 1 (g, K)- : 8

19 9 V = Cv k+2m v n r θ v n = e inθ v n Hv n = nv n m=0 X + v n Cv n+2 X v n Cv n 2 X v k = 0 D + k k 2 (holomorphic discrete series) k = 1 (limit of discrete series) L 2 (G) K (L 2 (G) K- ) v k lowest weight vector D + k [Kn, Chapter II, 5-6] X + X V = (anti-holomorphic discrete series) D k D + k m=1 Cv k 2m sl(2, C) lowest weight k k 0 f S k (Γ ) φ f L 2 (Γ \G) 3.1 φ f D + k vector lowest weight 3.2 S k (Γ ) Hom (g,k) (D k, L 2 (Γ \G)) f S k (Γ ) D k lowest weight vector φ f (g, K)- L 2 (Γ \G) G 3.1 X φ f = 0 U(g) g U(g) = 1 4 (H2 + 2X + X + 2X X + ) ( U(g) ) Casimir U(g) Z(g) Z(g) = C[ ] f M k (Γ ) ( k ( )) k φ f = 0 9

20 10 Z(g) = C[ ] Schur X G 3.2 G f (1) f Γ - γ Γ, g G f(γg) = f(g). (2) f K- g G dim f(gk) k K C < (3) f Z(g)- 0 p(x) p( )f = 0. (4) f C, M f(g) < C g M. A(Γ, K) f cuspidal f(ng) = 0 a. a. g, N f cusp cusp A 0 (Γ, K) (3) f U(g) (2) (3) f smooth (cf. [Bo, Theorem 2.13]). ψ L 2 (Γ \G) 3.1 (5) N ψ(ng) dn = 0 φ cuspidal cuspidal L 2 cusp(γ \G) G L 2 (Γ \G) K L 2 disc (Γ \G) K L 2 disc (Γ \G) K = V L 2 (Γ \G) K V : L 2 (Γ \G) L 2 disc (Γ \G) K L 2 (Γ \G) K V 10

21 (1) L 2 0 (Γ \G) (g, K)- L 2 0(Γ \G) K = m(π)v π, m(π) < (2) (g, K)- π: L 2 disc (Γ \G) K = L 2 0(Γ \G) K H res H res Eisenstein residue Γ = SL(2, Z) C [Bo, Theorem 16.6] SL(2, R) Hecke L- ( ) 1 0 Hecke 0 p SL(2) GL(2) G = GL(2), A = A Q Q A f = p< Q p, G Q = GL(2, Q) G = GL(2, R), G + = GL + (2, R) := {γ GL(2, R) det g > 0} G Af G A G - 1 K 0 G Af det : K f A ( ) G A = G Q G + K 0 (4.1) 4.1 Γ = K 0 SL(2, Q) Γ \H G Q R + \G A/K 0 Γ \SL(2, R) G A G Q 11

22 12 Example 4.2 N {( ) } a b K 0 (N) = GL(2, c d Ẑ) c 0 mod N = GL 2 (Z p ) K p p N p N ( K p = {( a c ) } b GL(2, Z p ) d c 0 mod p ) Γ = K 0 (N) SL(2, Q) = Γ 0 (N) ψ modn Dirichlet f M k (Γ 0 (N), ψ) ψ A ω ψ ω ψ : A A /Q R + = Z p p N Z p (Z/NZ) ψ C 1 ω ψ K 0 (N) ( ) a b ω ψ : ω ψ (d) C 1 c d 4.3 f M k (Γ 0 (N), ψ) G A Φ f (4.1) G A g = γg k 0 Φ f (g) = f(g (i))j(g, i) k ω ψ (k 0 ) 1 well-defined (i) γ G Q Φ(γg) = Φ(g) ( G Q - ) (ii) k 0 K 0 (N) Φ f (gk 0 ) = Φ(g)ω ψ (k 0 ) 1 (iii) Φ f (gr θ ) = e ikθ Φ f (g) (iv) Φ f G + smooth Φ f = k ( ) k Φ f 12

23 13 (v) z Z A A Φ f (zg) = ω ψ (z) 1 Φ(g) (vi) Φ f c > 0 G A Ω C N g Ω a > c a A (( ) ) a 0 Φ f g C a N 0 1 (vii) f S k (Γ 0 (N), ψ) (( ) ) 1 x Φ f g dx = Q/A a.a. g (ii), (iii) K = K K 0 (1)- (vi) Z(g)- ω Hecke ω : A /Q R + C1 L2 (G Q \G A, ω) G A 1. γ G Q Φ(γg) = Φ(g) 2. z Z A Φ(zg) = ω(z)φ(g) 3. Φ(g) 2 dg < Z A G Q \G A f S k (Γ 0 (N), ψ) Φ f L 2 (G Q \G A, ω 1 ψ ) 4.2 G A Hecke L 2 (G Q \G A, ω) G A G A G = GL(2, R) G p = GL(2, Q p ) G 4.3 (iii), (iv) 2 G p G p (g, K)- G p (π, V ) K p = GL(2, Z p ) V = σ K p V (σ), dim V (σ) < 13

24 14 V ( ) K- f ( f G p Hecke ) π(f)v = f(g)π(g)v dg G p (π, V ) L 2 (G Q \G A, ω) π(f)φ f(g) = f(g 1 ) Φ G p Hecke ( 4.4 ) Φ L 2 (G Q \G A ) ( ω ) T (p)φ Φ H p = p 0 K p K p 0 1 T (p)φ = Φ(gh) dh (4.2) H p f S k (Γ 0 (N)) p k/2 1 T (p)φf = Φ Tpf ) f S k (Γ 0 (N)) (4.2) g G A (4.1) g = γg k (γ G Q, g G, k K 0 (N)) Φ f G Q γ = 1 2 h k p- 1 2 p N (p N ) H p K p ( ) ( ) p 1 p b 1 0 H p = K p K p p b=0 ω = ω 1 ψ (4.2) ( )) ( ( )) p 1 p b 1 0 T (p)φ f (g) = Φ f (g + Φ f g p b=0 K p Haar vol(k p ) = 1 ( ) Φ f f p b G Q G K 0 (N) p ( ) G A g G A p- 1 2 p b γ = G Q 0 1 ( ) ( ) ( ) p b p b p 1 bp 1 g = g k p = γ g k k 0 14

25 15 ( ) 1 k 0 p- 1 p b 2 K 0 (N) 0 1 ( )) ( ) p b z + b Φ f (g = p k/2 1 f 0 1 p ( g (i) = z ) ( )) 1 0 Φ f (g = p k/2 f(pz) 0 p f S k (Γ 0 (N), ψ) L 2 (G Q \G A, ω 1 ψ ) G A R H p h (( )) a b h = ω ψ (d) c d R(h)Φ f Hecke L 2 0 (G Q\G A, ω) L 2 (G Q \G A, ω) cuspidal G A R 0 ω 4.5 ( 1 ) G A (R 0 ω, L 2 0 (G Q\G A, ω)) 1 Φ Fourier Whittaker Whittaker [Ko] Whittaker [AL] A.O.L. Atkin and J. Lehner Hecke operators on Γ 0 (m), Math. Ann. 185 (1970), p [Bo] A. Borel, Automorphic forms on SL 2 (R), Cambridge Tracts in Mathematics, 130. Cambridge University Press, Cambridge, (1997). [Bu] D. Bump, Automorphic forms and representations, Cambridge University Press, (1997). 15

26 16 [Ge] [JL] S.S.Gelbart. Automorphic forms on adele groups, Annals of Math.Studies. Princeton University Press and University of Tokyo Press, Princeton, N.J., 83, H. Jacquet, and R.P. Langlands, Automorphic forms on GL(2)., Lecture notes in Mathematics 114, (1970) Springer Verlag. [Kn] A.W. Knapp, Representation theory of semisimple groups, An overview based on examples. Princeton Mathematical Series, 36. Princeton University Press, Princeton, NJ, [Ko], GL2 L, 16 L (2009), p [Ma], Weil Howe duality [Mo], Hecke, 18 (2010), p

27 17 Shimura 1 Riemann zeta Theta Dedekind η, Hecke [14] Metaplectic Metaplectic Metaplectic [14] Koblitz[3] [4] 4, a Z, b 2Z + 1 (b 0) ( ) a b 2 (1) (a, b) 1, ( ) a b = 0. (2) b, ( ) a b. (3) b > 0 a ( ) a b b (4) a 0 b ( ) a b Q( ( a)/q ) (5) a > 0 (resp. a < 0) a = 1 (resp. 1). (6) ( 0 ±1) = N Dirichlet χ (n, N) > 1 χ(n) = 0 z, x e(z) = exp(2π 1z) z x = exp(x log(z)) log(z) π < arg(z) π µ( ) Möbius 1 H = {z C Im(z) > 0}, C 1 = {z C z = 1} 1

28 18 H GL + 2 (R) ; ( α(z) = az + b cz + d, α = a c ) b GL + 2 (R), z H. d N N {( ) a b Γ 0 (N) = SL 2 (Z) c d {( ) a b Γ 1 (N) = Γ 0 (N) c d {( ) a b Γ (N) = Γ 1 (N) c d c 0 (mod N) } } a d 1 (mod N), } b 0 (mod N) N Dirichlet χ Γ 0 (N) ( ) a b χ(γ) = χ(d), γ = Γ 0 (N) c d Γ = { ± ( } 1 n 0 1) n Z [SL2 (Z) : Γ 0 (M)] i M theta θ(z) = n Z e(n 2 z) z H, 2.2 k, Γ SL 2 (Z) χ : Γ C 1 H f (1),(2)( f ) k χ Γ a b (1) α = Γ (cz + d) c d k f(α(z)) = χ(α)f (2) f Γ cusp Γ cusp cusp k χ Γ cusp M k (Γ, χ), S k (Γ, χ). 2.3 Metaplectic k GL + 2 (R) G G = G(k + 1/2) { } = (α, φ(z)) α GL + 2 (R), φ(z) = t(det α) k/2 1/4 (cz + d) k+1/2 (t C 1 ). G (α 1, φ 1 (z)) (α 2, φ 2 (z)) = (α 1 α 2, φ 1 (α 2 (z))φ 2 (z)) G GL + 2 (R) Pr : G GL + 2 (R) Ker(Pr) = C 1 G GL+ 2 (R) C 1 2

29 19 G H f f ξ = φ(z) 1 f(α(z)), ξ = (α, φ(z)) G. G ξ 1, ξ 2 G f (ξ 1 ξ 2 ) = (f ξ 1 ) ξ 2 Definition 1. G Γ (1),(2),(3) Γ G Fuchs (1) Pr( Γ ) SL 2 (Z) (2) Pr : Γ Pr( Γ ) (3) 1 Pr( Γ ) Pr 1 ( 1) = {( 1, 1)}. Γ Example 1. 4 N G Fuchs j(α, z) = θ(α(z)), α Γ 0 (4), z H θ(z) j(α 1 α 2, z) = j(α 1, α 2 z)j(α 2, z) Poisson formula ( ) ( ) 1 1/2 ( c ) j(α, z) = (cz + d) 1/2 a b, α = Γ 0 (4) d d c d N even Dirichlet χ Γ 0 (N) γ (γ, χ(γ)j(γ, z) 2k+1 ) G χ( 1)j( 1, z) 2k+1 = 1 { ( ) } Γ 0 (N, χ) = (α, χ(d)j(α, z) 2k+1 a b ) α = Γ 0 (N) c d G Fuchs α Γ 0 (N) G α = (α, χ(d)j(α, z) 2k+1 ) Γ 0 (N, χ) Γ } (N) = {α α Γ (N) 2.4 Definition 2. k Γ G Fuchs H f (1),(2) f k + 1/2 Γ ( ) (1) ξ Γ f ξ = f. (2) f Pr( Γ ) cusp., Pr( Γ ) cusp cusp k + 1/2 Γ cusp M k+1/2 ( Γ ), S k+1/2 ( Γ ). 4 N M k+1/2 (N, χ) = M k+1/2 ( Γ 0 (N, χ)), S k+1/2 (N, χ) = S k+1/2 ( Γ 0 (N, χ)) S k+1/2 ( Γ ) Petersson 1 k+1/2 dxdy < f, g >= [SL 2 (Z) : Pr( Γ f(τ)g(τ)imτ )] Pr( Γ )\H y 2, f, g S k+1/2( Γ ) 3

30 20 Example 2. ψ r ν ψ( 1) = ( 1) ν 0 1 Theta h(z; ψ) = 1 ψ(m)m ν e(m 2 z) 2 m Z h(z; ψ) M ν+1/2 (4r 2, ψ ν ), ψ ν (d) = ψ(d) ( ) 1 ν 4r 2 d ψ = 1 h(z; 1) = 1 2 θ(z) M 1/2(4, 1) ψ(m) = ( 3 m) h(z; ψ) = η(24z) = e(z) n 1(1 e(24nz)) M 1/2 (4r 2, ψ) 2.5 Hecke Γ G Fuchs Pr( Γ ) = Γ Pr Γ Γ Γ Pr 1 Γ L : Γ Γ α GL + 2 (R) Γ αγ α 1 ξ G Pr(ξ) = α γ Γ α 1 Γ α Pr(L(αγα 1 )) = Pr(ξ L(γ)ξ 1 ) L(αγα 1 ) = ξl(γ)ξ 1 (1, t(γ)), γ Γ α 1 Γ α Ker(Pr) = {(1, t) t C 1 } G t(γ) ξ t : Γ α 1 Γ α C 1 Proposition 1. Γ ξ 1 Γ ξ = L(Ker(t)) [Γ : Ker(t)] < Γ ξ Γ ξ 1 Γ ξ Γ = ν: α [Γ : Ker(t)] < f M k+1/2 ( Γ ) f [ Γ ξ Γ ] = ν: ξ ν M k+1/2 ( Γ ) (1) t Γ = ν: Γ ξ ν f ξ ν (Γ α 1 Γ α)γ ν, Γ α 1 Γ α = µ: Ker(t)δ µ 4

31 21 Γ = ν L(Γ α 1 Γ α)l(γ ν ) = ν,µ L(Ker(t))L(δ µ γ ν ) = ν,µ( Γ ξ 1 Γ ξ)l(δµ γ ν ) Γ ξ Γ = ν,µ Γ ξ L(δ µ γ ν ) f [ Γ ξ Γ ] = ( ) f ξl(δ µ ) L(γ ν ) ν µ ( ) ( ) = t(δ µ ) f ξl(γ ν ) = 0. µ (2) t L(αγα 1 ) = ξl(γ)ξ 1 (γ Γ α 1 Γ α) Pr Γ ξ Γ Γ αγ Γ αγ = ν Γ α ν Γ ξ Γ = ν Γ ξ ν, Pr(ξ ν ) = α ν ξ ν Γ ξ Γ {ξ ν } {α ν } [ Γ ξ Γ ] Example 3. m, n N, t C 1, 4 N ( ) m 0 α =, ξ = 0 n m = m/(m, n), n = n/(m, n) ( a ( m ξγ ξ 1 = (αγα 1 ) (1, n d ν ( α, t (n/m) k/2+1/4) G )), γ = c ) b Γ 0 (4) α 1 Γ 0 (4)α d m n mn t(γ) = ( m n d ) f M k+1/2 (N, χ) f [ Γ 0 (N, χ)ξ Γ 0 (N, χ)] = 0, mn t 1, m = 1, n = p 2 (p ) f [ Γ 0 (N, χ)ξ Γ 0 (N, χ)] Γ 0 (N)\Γ 0 (N)αΓ 0 (N) G Γ 0 (N, χ)ξ Γ 0 (N, χ) (( ) ) 1 0 Γ 0 (N, χ) 0 p 2, p k+1/2 Γ 0 (N, χ) Γ 0 (N, χ) 1 m Γ0 (N, χ) p m Γ0 (N, χ) p2 0 m Z/p 2 Z 0 p 2 m (Z/pZ) 0 p 0 1 = (p, N) = 1 Γ 0 (N, χ) 1 m p N. 0 p 2 m Z/p 2 Z 5

32 22 ξ = (( ) ) p, p k+1/2 Γ 2 0 (N, χ) ( ) 1 m 0 p 2 ( ) p m 0 p ( ) p = = = = = (( ) ) ( ) (( ) ) p 2, p k+1/2 1 m 1 m = p 2, p k+1/2, ( ) (( ) ) ( ) Nps 1 0 p 2, p k+1/2 p m (pr + Nsm = 1) Ns r (( ) ( ) p m 1 (k+1/2) ( ) ) m, χ(p), 0 p p p ( ) (( ) ) ( ) p 2 t 1 0 N d 0 p 2, p k+1/2 p 2 d t (p 2 d + Nt = 1) N 1 (( ) ) p 2 0, p (k+1/2) χ(p 2 ). 0 1 f M k+1/2 (N, χ) [ (( ) ) ] 1 0 f Γ 0 (N, χ) 0 p 2, p k+1/2 Γ 0 (N, χ) f 1 m, p k+1/2 + f p m, χ(p) m Z/p 2 Z 0 p 2 m (Z/pZ) 0 p + f p2 0, p (k+1/2) χ(p 2 ) p N, = 0 1 m Z/p 2 Z f 1 m, p k+1/2 p N, 0 p 2 ( 1 p ) (k+1/2) ( ) m p Fourier Theorem Theorem 1. ([Shimura[14]:Theorem 1.7]) k Z, 4 N, p M k+1/2 (N, χ) p 2 -th Hecke [ (( ) ) ] T (p 2 ) := p k 3/2 1 0 Γ 0 (N, χ) 0 p 2, p k+1/2 Γ 0 (N, χ) f = n 0 a(n)e(nz) M k+1/2(n, χ) f T (p 2 ) = n 0 b(n)e(nz) Fourier b(n) ( ) n a(p 2 n) + χ 1 (p) p k 1 a(n) + χ(p 2 )p 2k 1 a(n/p 2 ) p N, b(n) = p a(p 2 n) p N, χ 1 (m) = χ(m) ( ) 1 k m N Dirichlet p2 n a(n/p 2 ) = 0 6

33 23 ( (( ) ) p Γ 0 (N) 0 p )Γ 2 0 (N) Γ 0 (N, χ) 0 p, p k+1/2 Γ0 (N, χ) 2 T (p 2 ) p T (p 2 ) M k+1/2 (N, χ) Theorem 2. ([Shimura[14]:Corollary 1.8]) p f T (p 2 ) = ω p f, ω p C f = n 0 a(n)e(nz) M k+1/2(n, χ) p N p 2 t (n, p) = 1 n ω p a(tn 2 ) = a(tn 2 p 2 ) + χ 1 (p) ( ) t p k 1 a(tn 2 ), p ω p a(tp 2m n 2 ) = a(tn 2 p 2m+2 ) + χ(p 2 )p 2k 1 a(tp 2m 2 n 2 ), (m 1) a(n) Fourier f p N χ χ(p) = 0 Theorem 2 T (p 2 ) (p ) f = n 0 a(n)e(nz) M k+1/2(n, χ) (N, t) = 1 t Dirichlet n 1 a(tn2 )n s Euler Dirichlet p a(tn 2 )n s = a(tn 2 p 2l )p sl n s = n 1 (n,p)=1 l 0 (n,p)=1 H n,p (x) = m 0 a(tp2m n 2 )x m H n,p (p s )n s ω p x H n,p (x) = ω p a(tn 2 )x + ω p a(tp 2m n 2 )x m+1 m 1 ( ( ) ) t = a(tn 2 p 2 ) + χ 1 (p) p k 1 a(tn 2 ) x p + ( ) a(tp 2m+2 n 2 ) + χ(p 2 )p 2k 1 a(tp 2m 2 n 2 ) x m+1 m 1 ( ) t = H n,p (x) a(tn 2 ) + χ 1 (p) p k 1 a(tn 2 )x + χ(p 2 )p 2k 1 x 2 H n,p (x) p ( H n,p (x) 1 ω p x + χ(p 2 )p 2k 1 x 2) ( ( ) ) t = a(tn 2 ) 1 χ 1 (p) p k 1 x p a(tn 2 )n s = n 1 (n,p)=1 a(tn 2 )n s ( 1 χ 1 (p) (1 ω p p s + χ(p 2 )p 2k 1 2s) 1 ( ) ) t p k 1 s p (n,p)=1 a(tn2 )n s 7

34 24 Theorem 3. ([Shimura[14]:Theorem 1.9]) t 1 N p f T (p 2 ) = ω p f, ω p C f = n 0 a(n)e(nz) M k+1/2(n, χ) Dirichlet n 1 a(tn2 )n s Euler a(tn 2 )n s = a(t) n 1 p: ( ( ) ) t ( 1 χ 1 (p) p k 1 s 1 ω p p s + χ(p 2 )p 2k 1 2s) 1. p Euler (1 ω p p s + χ(p 2 )p 2k 1 2s) 1 p: t M 2k (N, χ 2 ) Hecke Dirichlet Euler f S k+1/2 (N, χ) M 2k (N, χ 2 ) Hecke F ( Dirichlet ), f F ( Shimura ) 3 Shimura 3.1 Shimura f(z) S k+1/2 (N, χ) f(z) Theorem 3 Theorem 3 a(t) p: (1 ω p p s + χ(p 2 )p 2k 1 2s) 1 = = p: ( ( ) ) t 1 1 χ 1 (p) p k 1 s a(tn 2 )n s p n 1 m 1 χ 1 (m) ( ) t m m k 1 s a(tn 2 )n s n 1 = ( ) t χ 1 (d) a(tn 2 /d 2 )d k 1 n s d n 1 d n = A t (n)n s n 1 Dirichlet F t (z) = n 1 A t(n)e(nz) M 2k (N t, χ 2 ) ( N t t ) Weil 3.2 Shimura Weil ( [10] 4.5 ) 8

35 25 Theorem 4 (Weil). M Dirichlet φ Dirichlet D(s) = n 1 a(n)n s, (a n C) 3 n 1 a(n)e(nz) M 2k(M, φ) (1) D(s) Re(s) > σ (2) (a, b) = 1, b > 0 {a + bn n 0} P r {1} P (r, M) = 1 r ψ R(s, ψ) = (2π) s Γ (s) n 1 ψ(n)a(n)n s R(s, ψ) s σ 1 < Re(s) < σ 2 (3) Re(s) > σ Dirichlet n 1 b(n)n s (2) ψ ( ) ψ(m)φ(r)g(ψ) 2 R(2k s, ψ) = (r 2 M) s k ψ(n)b(n)n s r n 1 g(ψ) = r i=1 ψ(i)e(i/r) Remark 1. Theorem 4 Dirichlet D(s) Re(s) < 2k s n 1 a(n)e(nz) S 2k(M, φ) Shimura A t (n)n s = ( ) t χ 1 (d) a(tn 2 /d 2 )d k 1 n s d n 1 n 1 d n Theorem 4 3 F t (z) = n 1 A t(n)e(nz) M 2k (N t, χ 2 ) f(z) = n 1 a(n)e(nz) S k+1/2(n, χ) Theorem 3 (1) n 1 A t(n)n s. a(n) = O(n k/2+1/4 ) a(tn 2 )n s a(tn 2 ) n Re(s) const. t k/2+1/4 n (k+1/2) Re(s). n 1 n 1 n 1 n 1 a(tn2 )n s Re(s) > k+3/2 ( ) t m 1 χ 1(m) m k 1 s Re(s) > k m n 1 A t(n)n s Re(s) > k + 3/2 (2) 1 N r ψ D t (s, ψ) = n 1 ψ(n)a t (n)n s s. t t = 1 t = 1 D 1 (s, ψ) = ψ(n)a 1 (n)n s = ψ(m)χ 1 (m)m k 1 s ψ(m)a(m 2 )m s n 1 m 1 m 1 9

36 26 Dirichlet Rankin (4π) s Γ (s) m 1 ψ(m)a(m 2 )m s = = = = 0 Γ \H ( 1 0 ) f(z)h(z; ψ)dx y s 1 dy s+1 dxdy f(z)h(z; ψ)y Γ 0 (Nr 2 )\H γ Γ \Γ 0 (Nr 2 ) Γ 0 (Nr 2 )\H y 2 f(z)h(z; ψ)y s+1 s+1 dxdy f(γ(z))h(γ(z); ψ)im(γ(z)) (c,d)=1, c 0 (mod Nr 2 ) c>0 if c 0, d=1 if c=0 y 2 φ(d)(cz + d) k ν cz + d 2ν 1 2s dxdy y 2, φ(d) = χ(d)ψ(d) ( ) 1 k, d = = 2 (4π) s Γ (s) n 1 Γ 0 (Nr 2 )\H Γ 0 (Nr 2 )\H ψ(n)a 1 (n)n (2s ν) f(z)h(z; ψ)y s+1 φ(m)m k+ν 1 2s m 1 m.n Z (Nr 2 m,n)=1 f(z)h(z; ψ)y s+1 φ(n)(nr 2 mz + n) k ν Nr 2 mz + n 2ν 1 2s dxdy m.n Z (m,n) (0,0) φ(n)(nr 2 mz + n) k ν Nr 2 mz + n 2ν 1 2s dxdy y 2. y 2 C(z, s) = m.n Z (m,n) (0,0) φ(n)(nr 2 mz + n) k ν Nr 2 mz + n 2ν 1 2s Eisenstein φ Lemma 1. ([Shimura[14];Lemma 3.1]) A φ A α A = 1 α > 0 H α (s, z, φ) = π s Γ (s)y s φ(n)(amz + n) α Amz + n 2s m.n Z (m,n) (0,0) Re(s) > α s ( H α (α + 1 s, z, φ) = ( 1) α g(φ)a 3s α 2 z α H α s, 1 ) Az, φ, g(φ) = 10 A φ(k)e(k/a) k=1

37 27 α = k l 0, 4A B f S(k + 1/2, Γ (B)), g M(l + 1/2, Γ (B)) s C Γ (B)\H f(z)g(z)y l+1/2 H α (s, z, φ) dxdy y 2 s Lemma 1 φ φ 0 Eisenstein C(z, s) χ 1 (m) = χ(m) ( ) 1 k m M N = MK E = p C(z, s) = = m.n Z (m,n) (0,0) m.n Z (m,n) (0,0) = 0<l E 0<l (n,e) µ(l) φ 0 (n)(nr 2 mz + n) k ν Nr 2 mz + n 2ν 1 2s µ(l)φ 0 (l)l 0<l (n,e) n=ln,k=ll µ(l)φ 0 (l)l k+ν 1 2s, 2t = 2s 2ν + 1 p: p N,p M k+ν 1 2s φ 0 (n )(Ml r 2 mz + n ) k ν Ml r 2 mz + n 2ν 1 2s (m,n) (0,0) 2 (4π) s Γ (s)π t Γ (t)d 1 (2s ν, ψ) = (Kr) t µ(l)φ 0 (l)l k s 1/2 0<l E Γ 0 (Nr 2 )\H φ 0 (n)(ml r 2 mz + n) k ν Ml r 2 mz + n 2ν 1 2s. f(z)h(z; ψ)y ν+1/2 H k ν (t, (K/l)rz, φ 0 ) dxdy y 2 Lemma 1 R 1 (s, ψ) = (2π) s Γ (s)d 1 (s, ψ) s (3) (2) r ψ R t (s, ψ) = (2π) s Γ (s)d t (s, ψ) ( ψ(nt )χ 2 (r)g(ψ) 2 ) R t (2k s, ψ) = (r 2 N t ) s k (2π) s Γ (s) ψ(n)b n n s r n 1 N t Dirichlet D(s) = n 1 b nn s. t = 1 t = 1 s k + ν s 2 (4π) s Γ (s)π t Γ (t)d 1 (2s ν, ψ) f(z)h(z; ψ)y ν+1/2 H k ν (t, (K/l)rz, φ 0 ) dxdy y 2 (3.1) Γ 0 (Nr 2 )\H ( ) Dirichlet t = s ν +1/2 t + (k ν) + 1 Lemma 1 ( H k ν (k ν) + 1 t, Krz ), φ 0 l ( ) Krz k ν ( = ( 1) k ν g(φ 0 )(Mr) 3t (k ν) 2 H k ν t, l ) l Nr 2 z, φ 0 11

38 28 (3.1) z 1/Nr 2 z Atkin- Lehner = = ( ) ( ) ( ) ν+1/2 f Nr 2 h z Nr 2 z ; ψ Im Nr 2 z ( N k 2 + ( 1 4 1r 2 z ) k+ 1 2 ( g(r z)) 2 ) ν 1 r 1 2 g(ψ) 4 ( ( ) k+1 ) N k r 2k ν ( ) 4 g(ψ)( 1) ν+ 1 N 2 z k ν g(r 2 z)h 2 4 z; ψ g(z) = f Γ 0 (Nr 2 )\H ( 2rN 1z ) ν+ 1 ( ) ( ) 2 N 1 ν+1/2 h 4 z; ψ Im Nr 2 z y ν+ 1 2, ( ( 0 1 ) N 0, N k/2+1/4 ( 1z ) ) k+1/2 ( ( S k+1/2 N, χ N )), f(z)h(z; ψ)y ν+1/2 H k ν ( t + (k ν) + 1, (Krz/l), φ 0 ) dxdy = (elementary constant factor) Γ 0 (Nr 2 )\H y 2 ( ) N g(r 2 z)h 4 z; ψ y ν+ 1 2 Hk ν (t, (K/l)rz, φ 0 ) dxdy y 2 (3.1) (2) Dirichlet ( [14] ). (1), (2), (3) D t (s, ψ) Weil F t (z) = n 1 A t(n)e(nz) M 2k (N t, χ 2 ) Remark 2. Dirichlet D t (s, ψ) f S k+1/2 (N, χ) p cond(χ t ) N p T (p 2 ) D t (s, ψ) Theorem 5. ([Shimura[14];Main Theorem]) k, N N 4 N χ N ( even ) Dirichlet t χ t (m) = χ(m) ( ) 1 k ( t ) m m tn Dirichlet f = n 1 a(n)e(nz) S k+1/2(n, χ) p N, p cond(χ t ) p T (p 2 ) S t,χ k,n (f) = n 1 d n χ t (d)a(tn 2 /d 2 )d k 1 e(nz) N t M 2k (N t, χ 2 ) Hecke k 2 n 1 A t(n)n s 2k > Re(s) > k + 3/2 f S 2k (N t, χ 2 ) Hecke Theorem 5 N t (3) R t (2k s, ψ) M M (3) M N t 12

39 29 χ t (m) = χ(m) ( ) 1 k ( t ) m m Mt, t 2 = (t, N) χ (m) = χ(m) ( ) t 2m M ν(2) = 4, ν(3) = 2, ν(p) = 1 (p > 3) ν(n) H = p p N, p M t p ν(p), K 0 = p H (f T (p 2 ))/f 0 N = [N, H 2, M ] N t = N /2K 0 N t Best possible N t Best possible 4 Shimura 4.1 Theta Kernel Niwa, Kojima, Cipra Theta Kernel ( k = 1 ), N t N/2 ( S k+1/2 (N, χ) N t ) k = 1 cusp S k+1/2 (N, χ) ( ) N t Best possible Metaplectic Theta Kernel cusp Theta Kernel Zagier Doi-Naganuma Zagier N t Symplectic Weil unitary, Metaplectic pair theta lift (Howe duality 1 ), Lion-Vergne[8] Howe duality Zagier ([8] [1] [9] ) t = 1 S 1,χ k,n ( t Theorem Fourier Kohnen S t,χ k,n 5 ) H τ z τ = Re(τ) + iim(τ) = x + iy, z = Re(z) + iim(z) = ρ + iσ Theorem 6. (Zagier identity[19, 8]) k > 1, N N χ 4N Dirichlet Ω k,n,χ (τ, z) = ( 1) k n 1 (a,b,c) Z 3 4N 2 a,4n b b 2 4ac=16N 2 n χ(c)(az 2 + bz + c) k n k 1 2 e(nτ), (τ, z) H 2 13

40 30 f S k+1/2 (4N, χ) S 1,χ k,n (f)(z) = Γ 0 (4N)\H f(τ)ω k,n,χ (τ, z)(imτ) k+ 1 2 S 1,χ k,n (f)(z) S 2k(2N, χ 2 ) N 1 2N Proof. Riesz f(τ) = n 1 a(n)e(nτ) S k+1/2(4n, χ) a(n) (n-th Poincaré P k+ 1 2,4N,n,χ(τ) S k+1/2(4n, χ) ) ( ) Ω k,n,χ (τ, z) = i 4N (4π) k 1 2 Γ (k 1 2 ) n 1 < f, P k+ 1 2,4N,n,χ > = Γ (k 1 2 ) a(n) i 4N (4πn) k 1 2 n k 1 Ω k,n,χ (τ, z) H H < f, Ω k,n,χ (τ, z) > = i 4N (4π) k 1 2 Γ (k 1 2 ) n k 1 n 1 d n = n 1 d n = S 1,χ k,4n (f) dxdy y 2 χ 1 (d)(n/d) k P k+ 1 2,4N,(n/d)2,χ (τ) e(nz) d n χ 1 (d)a(n 2 /d 2 )d k 1 e(nz) S 1,χ k,4n (f)(z) = χ 1 (d)(n/d) k < f, P k+ 1 2,4N,(n/d)2,χ > e( nz) Ω k,n,χ (τ, z) = i 4NΩ k,n,χ (τ, z) Γ 0 (4N)\H f(τ)ω k,n,χ (τ, z)(imτ) k+ 1 2 Ω k,n,χ (τ, z) τ Fourier Zagier (Zagier Doi-Naganuma ([13] )) Ω k,n,χ (τ, z) = ( 1) k n 1 ω k (z; χ, n)n k 1 2 e(nτ) dxdy y 2 ω k (z; χ, n) = ( ) χ(q)q(z, 1) k z, Q(z, 1) = (z, 1)Q 1 Q L N,n L N,n Γ 0 (2N) Q Q g = t gqg, g Γ 0 (2N) 14

41 31 2 { ( ) } a b/2 L N,n = Q = b/2 c 4N 2 a, 4N b, b 2 4ac = 16N 2 n Q = ( ) ( a b/2 b/2 c χ(q) = χ(c) g = a b c d) Γ0 (2N) ω k (g(z); χ, n) = ( ( χ(q)(cz + d) 2k (z, 1) t z gqg 1 Q L N,n = χ(d) 2 (cz + d) 2k ω k (z; χ, n), S 1,χ k,n (f) S 2k(2N, χ 2 ) )) k Remark 3. [8],Theta Kernel Ω k,n,χ (τ, z) Poincaré i 1 4N Ω k,n,χ (τ, z) Remark 4. Shimura Fourier Shimura Hecke k 2, N N, p S t,χ k,n S k+1/2 (4N, χ) S 2k (2N, χ 2 ) T (p 2 ) T (p) S k+1/2 (4N, χ) S t,χ k,n S 2k (2N, χ 2 ) Remark 5. f Theorem 5 S 1,χ k,n (f) (Theorem 6) Mellin (2π) s Γ (s) n 1 A 1 n s = = 0 S 1,χ dσ k,n (f)(iσ)σs σ ( f(τ) Ω k,n,χ (τ, iσ)σ s dσ ) k+1/2 dxdy y 0 σ y 2 (4.1) Γ 0 (4N)\H t = s + 1/ s (2π) s Γ (s) ( ) A 1 n s = f(τ) h(z; 1)C (z, t) y k+ 1 dxdy 2 y 2, (4.2) n 1 C (z, t) C (z, t) = (Kr) t 0<l E Γ 0 (4N)\H µ(l)φ 0 (l)l k t y k H k (t, (K/l)rz, φ 0 ) Eisenstein (4.1), (4.2) f Shimura L 2 s 2 h(z; 1)C (z, t) 2 0 Ω k,n,χ (τ, iσ)σ s dσ σ 15

42 Shintani Shimura S 1,χ k,n Shimura ( Shintani ) ( Shintani [16] [11] ) k > 1, N N g(z) = n 1 b(n)e(nz) S 2k(2N, χ 2 ) S 1,χ k,4n (g)(τ) = Γ 0 (2N)\H 2k dρdσ g(z)ω k,n,χ (τ, z)im(z) σ 2 Poincaré S 1,χ k,4n (g) S k+1/2(4n, χ) S 1,χ k,n S1,χ k,n Petersson S 1,χ k,4n (g) Fourier 2 Γ 1 (2N) Q = {γ Γ 1 (2N) Q γ = Q} ω k (z; χ, n) ω k (z; χ, n) = [Q] L N,n /Γ 1 (2N) γ Γ 1 (2N) Q \Γ 1 (2N) S 1,χ k,4n S 1,χ k,4n (g)(τ) = i N,k n 1 n k 1/2 [Q] L N,n /Γ 1 (2N) i N,k = ( 1) k [Γ 0 (2N)/{±1} : {±1}Γ 1 (2N)/{±1}] χ(q γ)(q γ)(z, 1) k ( ) g(z)χ(q)q(z, 1) k 2k dρdσ σ Γ 1 (2N) Q \H σ 2 e(nτ), ( ) g(z)χ(q)q(z, 1) k 2k dρdσ σ Γ 1 (2N) Q \H σ 2 = 2 2k+3 χ(q)π(16n 2 n) 1 2 k 2k 3 g(z)q(z, 1) k 1 dz. k 1 C Q C Q Q(z, 1) = 0 (1) a > 0 ( b 4N n)/2a ( b + 4N n)/2a a z 2 + brez + c = 0 Γ 1 (2N)\H (a < 0 ) (2) a = 0, b > 0 c/b i brez + c = 0 Γ 1 (2N)\H (b < 0.) g L ([6] ) Theorem 7. ([Kojima[6] ; Theorem 2]) g(z) = n 1 b(n)e(nz) S 2k(N, χ 2 ) Hecke S 1,χ k,4n (g)(τ) = d N,k n 1 ψ 0 j 2k 2 d N,k = ( 1) k 2 2k+3 (16N 2 ) 1 2 k π ( 2k 3 k 1 c j (ψ, g)l(g, j + 1, ψ) e(nτ), ), L(g, s, ψ) = n 1 ψ(n)b(n)n s Remark 6. Oda[12] (Niwa, Shintani, Zagier ) Shintani ( Zagier identity) (2, n 2) cusp [17] 16

43 Kohnen Shimura S k+1/2 (4N, χ) N t N/2 Shimura S t,χ k,4n (Kohnen ) N t k 2 N χ N Dirichlet N 1 χ χ pri χ χ( 1) = ϵ, χ = ( 4ϵ ) χ, N 2 = N/N 1 ξ k+1/2,ϵ = (( ), ϵ k+1/2 e ((2k + 1)/8) ) G Q k+1/2,4n,χ = [ Γ 0 (4N, χ )ξ k+1/2,ϵ Γ0 (4N, χ )] Q k+1/2,n,χ S k+1/2 (4N, χ ) hermitian Q k+1/2,n,χ S k+1/2 (4N, χ ) α = ( 1) [(k+1)/2] 2 2ϵ S + k+1/2 (N, χ) Kohnen Kohnen S + k+1/2 (N, χ) = f(z) = a(n)e(nz) S k+1/2 (4N, χ ) n 1 a(n) = 0 if ϵ( 1)k n 2, 3 (4) ϵ( 1) k D > 0, (N, D) = 1 D S + k+1/2 (N, χ) Shimura S D,χ k,4n H H Ω+ k,n,χ (τ, z) S D,χ k,4n (f)(z) = ( ) D χ(d)a( D n 2 /d 2 )d k 1 e(nz) d n 1 d n ( ) Ω + k,n,χ (τ, z) = i 4N (4π D ) k 1 2 Γ (k 1 2 ) 2 = ( i N (4π D ) k 1 2 Γ (k 1 2 ) = < f(τ), Ω + k,n,χ (τ, z) >, ) n 1 3 n 1 n k 1 n k 1 ( ( Pr S k+1/2 4N, 4ϵ ) ) χ S + k+1/2 (N, χ) ( ) D χ(d)(n/d) k Pr(P d k+ 1 2,4N,(n/d)2,χ (τ)) e(nz) d n ( ) D χ(d)(n/d) k Pr(P d k+ 1 2,4N,(n/d)2,χ (τ)) e(nz). d n Pr = 1 ( Qk+1/2,N,χ β ), β = α α β 2 τ Fourier Zagier Poincaré Fourier Proposition 2. ([Kohnen[5] ; Theorem 1], [Kojima-Tokuno[7] ; Theorem 2.2]) k 2 Ω + k,n,χ (τ, z) = i NC 1 k,d,χ pri n k 1 2 ( ) D µ(t) χ pri (t)t k 1 ω t k,n 2 1,N 2 /t (tz; D, ϵ( 1)k n, χ pri ) e(nτ). t N2 n 1 ϵ( 1) k n 0,1 (4) 17

44 34 C k,d,χpri χ pri Gauss W (χ pri ) ω k,n 2 1,N 2 /t 4.1 Zagier identity ω k (z; χ, n) ( ) ( ) C k,d,χpri = ( 1) [k/2] D k+1/2 2k 2 π 2 3k+2 χ pri ( D)W (χ pri )ϵ k 1 DN1 1/2 N1 k, ω k,n 2 1,N 2 /t (z; D, ϵ( 1)k n, χ pri ) = ω D (a, b, c)χ pri (c)(az 2 + bz + c) k. (a,b,c) Z 3 b 2 4ac=ϵ( 1) k N 2 1 Dn (N 2 1 N 2 /t) a ω D (a, b, c) Ω + k,n,χ (τ, z) τ Fourier ( D ) 2 D [a, b, c](x, Y ) D (b 2 4ac) 2 ( ) D (a, b, c, D) = 1 r [a, b, c](x, Y ), ω D (a, b, c) = r 0 (a, b, c, D) > 1 ω D (a, b, c) Γ 0 (1) r Proposition 2 Poincaré (τ ) Fourier ω k,n 2 1,N 2 /t (z; D, ϵ( 1)k n, χ pri ) ω D (a, b, c)χ pri (c)(az 2 + bz + c) k D N ( χ ) primitive 2 [a, b, c], 2 [at 2, bt, c] (t N 2 ) ω k,n 2 1,N 2 /t (z; D, ϵ( 1)k n, χ pri ) 2 ϵ( 1) k n 0, 1 (4) ω k,n 2 1,N 2 /t (z; D, ϵ( 1)k n, χ pri ) ω k,n 2 1,N 2 /t (z; D, ϵ( 1)k n, χ pri ) ( ) 2 ) S 2k (N 1 t, χ 2 pri S D,χ k,n (f)(z) S 2k(N, χ 2 ) S D,χ k,4n S + k+1/2 (N,χ) N D 4N/2 4N/4 D Zagier identity Kohnen 2 Remark 7. Kohnen S + k+1/2 (N, χ) Hecke T (p 2 ) (p 2 ) T (4) T (4) T (4) = 3 T 2 (4) Pr S + k+1/2 (N, χ) T (4) T (4) T (p 2 ) Kohnen 4 Hecke, T (4) T (p 2 ) (p 2), T (4) Shimura 4.4 Eisenstein Shimura Fourier Eisenstein Shimura Γ 0 (4) cusp i, 0 k + 1/2( 5/2) Eisenstein Ek+1/2 i (z) = j(α, z) 2k 1 M k+1/2 (4, 1), α Γ \Γ 0 (4) E 0 k+1/2 = ( 1) k i(z) k 1/2 E i k+1/2 H k+1/2 (z) = ζ(1 2k) = n 0 H(k; n)e(nz) ( 1 ) M 4z k+1/2 (4, 1) ( ( ) ) Ek+1/2 i (z) + 2 2k 1 1 ( 1) k i Ek+1/2 0 (z) 18

45 35 Eisenstein Fourier ζ(1 2k) if n = 0 ( ( )) 1 H(k; n) = 2 L dn ( ) ( ) dn 1 k, µ(d) d k 1 fn σ 2k 1 if n 1, n 0, 1 (mod 4) d d d f n 0 if n 1, n 2, 3 (mod 4) d n Q(( 1) k n)/q f n = n d n 1 Fourier H k+1/2 (z) M + k+1/2 (1, 1) Fourier µ n 1, D ( ) ) D d k 1 H (k ; n2 d d 2 D = 1 ( )) D (1 2 L ( ) D ( n ) k, µ(l) (ld) k 1 σ 2k 1 ld ld d n d n l (nf D /d) = 1 ( )) D (1 2 L k, σ 2k 1 (n) Eisenstein E 2k (z) = ζ(1 2k) σ 2k 1 (n)e(nz) M 2k (1, 1) n 1 Fourier σ 2k 1 (n) M + k+1/2 (1, 1) = CH k+1/2 S + k+1/2 (1, 1) M 2k(1, 1) = CE 2k S 2k (1, 1) S D,1 k,4 : M + k+1/2 (1, 1) M 2k(1, 1) a(n)e(nz) n 0 ( ( )) a(0) D 2 L 1 k, + ( ) ( D n d k 1 2 ) D a d d 2 e(nz), n 1 d n D ( 1) k D > 0 D,1 S k,4 S+ k+1/2 (1, 1) SD,1 k,4 S D,1 k,4 S + (1,1) = SD,1 k,4 : S+ k+1/2 k+1/2 (1, 1) S 2k(1, 1). Eisenstein S D,1 k,4 (H k+1/2) = 1 ( )) D (1 2 L k, ζ(1 2k)E 2k. D,1 Remark 8. k = 1 S 1,4 M + D,1 3/2 (1, 1) S 1,4 (M + 3/2 (1, 1)) = M 2 (1) S + D,1 3/2 (1, 1) S 1,4 Eisenstein ([2] ) k = 0 M + D,1 1/2 (1, 1) S 0,4 S D,1 0,4 (M + 1/2 (1, 1)) = M 0(1) M + 1/2 (1, 1) theta θ(z) = n Z e(n2 z) D,1 D 1 a(n) = δ(n = ) S k,4 (θ) = 1 2 L(1 k, ( D )) M 0 (1) = C 19

46 36 (1) ; 4, (1996). (2) B. A. Cipra; On the Niwa-Shintani theta-kernel lifting of modular forms, Nagoya Math.J. 91, (1983). (3) N.Koblitz; Introduction to Elliptic Curves and Modular Forms Graduate text in Math. 97, Springer- Verlag (1993). (4) N. ; ( 2 ) Springer-Japan (2008). (5) W. Kohnen; Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271, (1985). (6) H. Kojima; Fourier coefficients of modular forms of half integral weight, periods of modular forms and the special values of zeta functions, Hiroshima Math. J. 27, (1997). (7) H. Kojima-Y. Tokuno; On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen s spaces and the critical values of zeta functions, Tohoku Math. J. 56, (2004). (8) G.Lion-M.Vergne; The Weil representation, Maslov index and Theta series. Progress in Math. 6, Birkhäuser (1980). (9) ; 19 Weil Howe duality (2011). (10) T.Miyake; Modular Forms, Springer-Verlag (1989) (11) ; 8, (2000). (12) T. Oda; On Modular Forms Associated with Indefinite Quadratic Forms of Signature (2, n 2) Math.Ann. 231, (1977). (13) ; 8, (2000). (14) G.Shimura; On modular forms of half-integral weight, Ann. of Math. 97, (1973). (15) G.Shimura; On the Fourier coefficients of Hilbert modular forms of half-integral weight, Duke Math.J. 71, (1993). (16) T. Shintani; On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58, (1975). (17) ; 19 Oda lift (2011). (18) ; 8, (2000). (19) D. Zagier; Modular forms associated to real quadratic fields, Inventiones Math. 30, 1-46 (1975). 20

47 37 Accidental 1. ([Sak], ) ( ) ([Mat] ). dual pair.. Lie, SL(2, R) (SL(2, R) ) SL(2, R), dual pair SL(2, R) SL(2, R)., SL(2, R) dual pair O(p, q). Lie Accidental., SL(2, R). SL(2, R)/{±1} O 0 (2, 1). O 0 (2, 1) (2+, 1 ). SL(2, R) O(2, 1). Lie Accidental,.,, -, -,, Accidental. Sp(2; R)/{±1} O 0 (2, 3), (SL(2, R) SL(2, R))/{±( )} O 0 (2, 2), Lie Lie Dynkin Vogan. Helgason [He] Lie Accidental, Lie Accidental. Accidental, Lie. 1

48 38 Dynkin Vogan. 2 Lie, Lie, Lie Lie, Lie Lie Dynkin Humphrey[Hu]. Dynkin Lie Lie, Lie Lie, Vogan. 2.1, Lie Lie,. (1),, R, C Hamilton H. H {1, i, j, k}, {i, j, k}. (2) i 2 = j 2 = k 2 = 1, ij = ji = k. R i 2 = 1 i R + Ri C.. H {1, i, j, k }, {i, j, k }. i 2 = k 2 = 1, j 2 = 1, i j = j i = k.. C R R, H M(2, R). M(2, R). 2

49 39 (3) K (1) (2). K K x x K. K = R, C. C x + yi x yi C (K = C ), H x = x 0 + x 1 i + x 2 j + x 3 k x := x 0 x 1 i x 2 j x 3 k (K = H), H x = x 0 + x 1i + x 2j + x 3k x = x 0 x 1i x 2j x 3k (K = H ). H M(2, R) ( a M(2, R) X = c ) ( b ι d X := d c ) b M(2, R) a. M(n, K). K = R, C, C K Tr det. K = H, H M(n, K) M(n, K) C M(2n, C) R M(n, K). M(n, K),, τk n, N K n. n = 1 τ K, N K.. τ K (x) = x + x, N K (x) = x x (x K) X M(n, K), X X. 2.2 ([Hu, Chap.III. Section 9, 10] ) E, (α, β) (α, β E) E Φ,. (1) Φ, E. 3

50 40 (2) α Φ Φ ±α. (3) α Φ σ α GL(E)(reflection) Φ. 2(β, α) (4) α, β Φ α, β := (α, α) Z. σ α. σ α (β) := β α, β α (β E). β = α α, α. Φ. (1) E. (2) β Φ, {k α α } β = α k αα., = dim E. Φ, Φ Φ 1, Φ 2. Φ 1 Φ 2. α Φ i (α, β) = 0 β Φ j (i j, i, j {1, 2}).. Φ, Φ 1, Φ 2,, Φ m. Φ = Φ 1 Φ 2 Φ m. 2.3 Dynkin ([Hu, Chap.III. Section 11] ) Φ, Dynkin... α 1, α 2 α 1, α 2 α 2, α 1., (, ),. 4

51 41 Dynkin Dynkin. A n, B n, C n, D n, E 6, E 7, E 8, F 4, G 2 A n, B n, C n, D n, E 6, E 7, E 8, F 4, G 2. Lie Dynkin. g Lie. Lie g Cartan, g h Φ h (h h ) g h h. Φ h R g Killing. E = h R, Φ Lie g, Lie, Φ. Lie g Lie g = g 1 g 2 g m Φ Φ = Φ 1 Φ 2 Φ m. 1 i m Φ i Lie g i. Lie Lie Dynkin. Accidental Lie. Lie Lie Dynkin. 2.4 Dynkin Lie Lie, Dynkin.. Lie. 1. A n - SL(n, C) (sl(n, C)) Dynkin A-I SL(l, R) (sl(l, R)), A-II SL(m, H) (sl(m, H)), A III SU(p, q) (su(p, q)). 5

52 42 2. B n - D n - SO(2n + 1, C), SO(2n, C) (so(m, C)) Dynkin, B Dynkin, D BD-I BD-II SO(p, q) (so(p, q)), D-III SO (2n) (so (2n)). 3. C n - (c n - ) Sp(n, C) (sp(n, C)) Dynkin C-I Sp(n, R) (sp(n, R)), C-II Sp(p, q) (sp(p, q)). 2.5 Lie SL(l, R) := {g M(l, R) det(g) = 1}, SL(m, H) := {g M(m, H) NH m (g) = 1}, ( ) ( ) SU(p, q) := {g SL(p + q, C) t 1 p 0 p,q 1 p 0 p,q ḡ g = 0 q,p 1 q 0 q,p 1 q ( ) }, SU (2n) := { z 1 z 2 SL(2n, C) z 1, z 2 M(n, C)}( SL(n, H)), z 2 z 1 ( ) ( ) SO(p, q) := {g SL(p + q, R) t 1 p 0 p,q 1 p 0 p,q g g = 0 q,p 1 q 0 q,p 1 q ( ) ( ) }, SO (2n) := {g SU(n, n) t g 0 n 1 n 0 n 1 n g = }, 1 n 0 n 1 n 0 n ( ) ( ) Sp(n, R) := {g SL(2n, R) t g 0 n 1 n 0 n 1 n g = }, 1 n 0 n 1 n 0 n ( ) ( ) Sp(p, q) := {g M(p + q, H) t ḡ 1 p 0 p,q 1 p 0 p,q g = 0 q,p 1 q 0 q,p 1 q }. 6

53 43 Sp(p, 0) = Sp(0, p) Sp(p). Lie sl(l, R) := {X M(l, R) Tr(X) = 0}, sl(m, H) := {X M(m, H) τh m (X) = 0}, ( ) ( ) su(p, q) := {X sl(p + q, C) t 1 p 0 p,q 1 p 0 p,q X + 0 q,p 1 q 0 q,p 1 q ( ) X = 0 p+q }, su (2n) := { z 1 z 2 sl(2n, C) z 1, z 2 M(n, C)}( sl(n, H)), z 2 z 1 ( ) ( ) so(p, q) := {X sl(p + q, R) t 1 p 0 p,q 1 p 0 p,q X + 0 q,p 1 q 0 q,p 1 q ( ) ( ) X = 0 p+q }, so (2n) := {X su(n, n) t X 0 n 1 n 0 n 1 n + X = 0 2n }, 1 n 0 n 1 n 0 n ( ) ( ) sp(n, R) := {X sl(2n, R) t X 0 n 1 n 0 n 1 n + X = 0 2n }, 1 n 0 n 1 n 0 n ( ) ( ) sp(p, q) := {X M(p + q, H) t 1 p 0 p,q 1 p 0 p,q X + 0 q,p 1 q 0 q,p 1 q X = 0 p+q }. sp(p, 0) = sp(0, p) sp(p). 2.6 Vogan (1) Vogan ([Kn, Chap.VI, Section 8] ) g 0 Lie. θ g 0 Cartan. θ g 0 k 0 := {X g 0 θ(x) = X} g 0 Lie Lie. θ g 0 g 0 = k 0 p 0 (p 0 := {X g 0 θ(x) = X}) Cartan. Vogan θ Cartan h 0, h 0 k 0 h 0 k 0 θ Cartan h 0. g 0 Vogan, g := g 0 C Dynkin. ( h 0,C p 0,C ), ( h 0,C k 0,C ). 7

54 44 θ, ([Sat, 3, 3.4] ). Vogan, Cartan. (2) Vogan ([Kn, Appendix C, Section 3] ) Dynkin Lie Vogan A 2n sl(2n + 1, R) A 2n 1 sl(2n, R) A 2n 1 sl(n, H) (n 2) A p+q 1 su(p, q) (1 p q) p B p+q so(2p, 2q + 1) (1 p q) p B p+q so(2p, 2q + 1) (p > q 0) p C p+q sp(p, q) (1 p q) p C n sp(n, R) n D p+q+1 so(2p + 1, 2q + 1) (0 p q) p D p+q so(2p, 2q) (1 p q) p D n so (2n) (n 3) n Lie su(q) (A q 1 ), so(2q + 1) (B q ), sp(q) (C q ) so(2q) (D q ). B p+q. Vogan, Cayley. so(2p + 1, 2q + 1) (0 p q), so(1, 1), so(1, 3), p = 0. so(2p, 2q) (1 p q) so(2, 2). 8

55 45 3 Lie (Accidental ), Helgason[He] Lie. Lie, Dynkin. Accidental. Vogan B 2 = C 2 Lie Accidental ( [He, Chap.X, Section 6, 4]) A 1 = B 1 = C 1 1. su(2) so(3) sp(1). 2. sl(2, R) su(1, 1) so(2, 1) sp(1, R). B 2 = C 2 1. so(5) sp(2). 2. so(3, 2) sp(2, R). 3. so(4, 1) sp(1, 1). A 3 = D 3 1. su(4) so(6). 2. sl(4, R) so(3, 3). 3. su (4) sl(2, H) so(5, 1). 4. su(2, 2) so(4, 2). 5. su(3, 1) so (6). D 2 = A 1 A 1 1. so(4) so(3) so(3) su(2) su(2) sp(1) sp(1). 2. so(3, 1) sl(2, C). 9

56 46 3. so(2, 2) sl(2, R) sl(2, R). 4. so (4) su(2) sl(2, R). 1. so (8) so(6, 2). 3.2 Vogan Accidental (B 2 = C 2 ) Accidental B 2 = C 2.. so(5) sp(2), so(3, 2) sp(2, R), so(4, 1) sp(1, 1). C 2 Lie Vogan. sp(2) sp(2, R) sp(1, 1) B 2 Lie Vogan. so(5) so(2, 3) so(4, 1) C 2 B 2. B 2 = C 2 Accidental Vogan.. 4 Lie Accidental, Lie Accidental, Lie Accidental ( ).. [Yk, ]. Lie G, G 0 G. 10

57 Lie (Accidental ) A 1 = B 1 = C 1 1. SU(2) Sp(1) Spin(3), Sp(1)/{±1} SO(3). 2. SU(1, 1) SL(2, R). 3. SL(2, R)/{±1 2 } SO 0 (2, 1). B 2 = C 2 1. Sp(2) Spin(5), Sp(2)/{±1} SO(5). 2. Sp(2, R)/{±1 4 } SO 0 (2, 3). 3. Sp(1, 1)/{±1 2 } SO 0 (4, 1). A 3 = D 3 1. SU(4)/{±1 4 } SO(6). 2. SL(4, R)/{±1 4 } SO 0 (3, 3). 3. SU (4) SL(2, H), SL(2, H)/{±1 2 } SO 0 (5, 1). 4. SU(2, 2)/{±1 4 } SO 0 (4, 2). 5. SU(3, 1)/{±1 4 } SO (6). D 2 = A 1 A 1 1. SO(4) (Sp(1) Sp(1))/{±(1, 1)}. 2. SO 0 (3, 1) SL(2, C)/{±1 2 }. 3. SO 0 (2, 2) (SL(2, R) SL(2, R))/{±(1 2, 1 2 )}. 4. SO (4) (Sp(1) SL(2, R))/{±(1, 1 2 )}. I ( ) 1. SO (8)/{±1} SO 0 (6, 2)/{±1 8 }. 11

58 48 II ( Lie ) 1. SL(2, C)/{±1 2 } SO(3, C). 2. (SL(2, C) SL(2, C))/{±(1 2, 1 2 )} SO(4, C). 3. Sp(2, C)/{±1 4 } SO(5, C). 4. SL(4, C)/{±1 4 } SO(6, C). III (similitude ) 1. P GL(2, R)(:= GL(2, R)/R ) SO(2, 1). 2. P GSp(2, R)(:= GSp(2, R)/R ) SO(2, 3). 3. P GSp(1, 1)(:= GSp(1, 1)/R ) SO(4, 1). 4. P GL(2, H)(:= GL(2, H)/R ) SO(5, 1). 5. (GL(4, R) R )/{(z, z 2 ) z R } GSO(3, 3). 6. (GL(2, H) R )/{(z, z 2 ) z R } GSO(5, 1). 7. (GSp(1) GSp(1))/{(z, z 1 ) z R } GSO(4). 8. (GSp(1) GL(2, R))/{(z, z 1 ) z R } GSO (4). m Q similitude GO(Q), GSO(Q). GO(Q) := {g GL(m, R) t gqg = ν(g)q, ν(g) R }, GSO(Q) := {g GO(Q) det(g) = ν(g) m 2 }. 4.2, Accidental SL(2, R)/{±1 2 } SO 0 (2, 1), Sp(2, R)/{±1 4 } SO 0 (2, 3) 12

59 49, G G Lie, f : G G.. (i) G. (ii) Ker f. (iii) g, g G, G Lie dim g = dim g., C -Lie. G/ Ker f G. f df Lie df : g g. (ii) dim Ker df = 0 df, (iii) dim g = dim g, df. G G g. f. G/ Ker f G.. (V, Q) Q V. O(V, Q) := {g GL(V ) Q(gv) = Q(v), v V } (V, Q).. (1) (V, Q) (V 1, Q 1 ) := ({X M(2, R) t X = X}, det), ( ) (V 2, Q 2 ) := ({T M(4, R) T, Tr T = 0}, Tr) Tr V 2. V 2 X Tr(X 2 ) O(V 1, Q 1 ) O(2, 1), O(V 2, Q 2 ) O(2, 3). 13

60 50 (2) f : G O(V, Q) (G = SL(2, R), Sp(2, R)) G = SL(2, R), Sp(2, R) f : G O(V i, Q i ) (i = 1, 2). { gv t g (i = 1, g SL(2, R), v V 1 ) f(g)v := gvg 1 (i = 2, g Sp(2, R), v V 2 ). SL(2, R), Sp(2, R)., Lie ([Kn, p117], [Yk, p61] ) SL(2, R) Sp(2, R) SO(2) Sp(2, R) O(4) U(2). f. (3) 4.2 Ker f = {±1}. Lie sl(2, R) so(2, 1), sp(2, R) so(2, 3) 4.2, Lie. 4.3, (V, Q), f Sp(1)/{±1} SO(3): (1) (V, Q) : (2) f : V := {x H x + x = 0}, Q := N H. f(g)v := gvḡ (g Sp(1), v V ). Sp(2)/{±1} SO(5): (1) (V, Q) : (2) f : ( ξ V := { x ) x ξ R, x H}, Q := N 2 ξ H. f(g)v := gv t ḡ (g Sp(2), v V ). 14

61 51 Sp(1, 1)/{±1} SO 0 (4, 1): (1) (V, Q) : V := {X M 2 (H) t X = IXI, τ 2 H (X) = 0}, Q := NH. 2 ( ) 1 0 I :=. 0 1 (2) f : f(g)v := g 1 vg (g Sp(1, 1), v V ). SU(4)/{±1 4 } SO(6): (1) (V, Q) : ( ) ( ) ( X Y 0 ξ b V := { t Y X X =, Y = ξ 0 ā ) a, a, b, ξ C}, Q := N. b v V, N(v) := 1 4 Tr(vt v). (2) f : f(g)v := gv t g (g SU(4), v V ). SU(2, 2)/{±1 4 } SO 0 (4, 2): (1) (V, Q) : ( ) ( ) ( X Y 0 ξ b V := { X =, Y = t Y t X ξ 0 ā v V, N(v) := 1 4 Tr(I 2vI 2t v) (I 2 := (2) f : f(g)v = gv t g (g SU(2, 2), v V ). ) a, a, b, ξ C}, Q := N. b ( ) ) SL(4, R)/{±1 4 } SO 0 (3, 3): C, SL(4, R) SU(4, C )(C ), SL(4, R) SU(4, C ) 15

62 52. (1) (V, Q) : ( X V := { t Y ) ( ) ( Y 0 ξ b X =, Y = t X ξ 0 ā ) a, a, b, ξ C }, Q := N. b v V N(v) := 1 4 Tr(vt v). (2) f : f(g)v = gv t g (g SU(4,, C ) SL(4, R), v V ). (Sp(1) Sp(1))/{±(1, 1)} SO(4): (1) (V, Q) : V := H, Q := N H. (2) f : f((g 1, g 2 ))v := g 1 vḡ 2 ((g 1, g 2 ) Sp(1) Sp(1), v V ). SL(2, C)/{±1 2 } SO 0 (3, 1): (1) (V, Q) : V := {X M(2, C) t X = X}, Q := det. (2) f : f(g)v = gx t ḡ (g SL(2, C), v V ). (SL(2, R) SL(2, R))/{±(1 2, 1 2 )} SO 0 (2, 2): (1) (V, Q) : V := M(2, R), Q = det. (2) f : f((g 1, g 2 ))v := g 1 v t g ((g 1, g 2 ) SL(2, R) SL(2, R), v V ). (Sp(1) SL(2, R))/{±(1, 1 2 )} SO (4): H R H C. Sp(1) 16

63 53 {x H N H (x) = 1}, SL(2, R) Sp(1, N H ) := {x H N H (x) = 1}. SO (4). ( ) ( ) ( ) J 0 2 {X SL(4, C) X = 0 2 J X J 0 2, t 0 1 XX = 1 4 }. (J = ) 0 2 J 1 0 (1) (V, Q) : V = H C, Q = N H,C. N H,C N H H, C H C. (2) f : f((g 1, g 2 ))v := g 1 vḡ 2 ((g 1, g 2 ) Sp(1) Sp(1, H ), v V ). H H C ( M(2, C)), g 1 {x H N(x) = 1} = Sp(1) v H C. 4.4 SO (8)/{±1 8 } SO 0 (6, 2)/{±1 8 } : [F-H, Section 3]. SO (4n). ( ) ( ) {g M 2n (H) t 0 n 1 n 0 n 1 n ḡ g = }. 1 n 0 n 1 n 0 n n 3 n. [F-H] SO (8)/{±1 8 }, 2 ([Kr, Theorem 1.8] ). Lie Accidental, 4.2 ([Yk, 10 ] ). 3.1 Dynkin Lie Accidental. similitude, f. Lie. similitude, Hecke, similitude. 17

64 54 Clifford [E-G-M]. Vahlen Clifford Lie, Spin. Clifford Vahlen ([E-G-M, Section 6] ), Accidental. 4.2, 4.3 SL(2, H)/{±1} SO 0 (5, 1), SU(3, 1)/{±1 4 } SO (6)., Accidental.,. (i) E GSO(E, N E/F ) (E/F, N E/F ). (ii) (B B /{(z, z 1 ) z GL(1)} GSO(B, N B ) (B, N B ). (iii) P GL(2) SO(2, 1). (iv) P GSp(2) SO(2, 3). (v) P GSp(1, 1) SO(4, 1). (vi) P GL(2, B) SO(5, 1) (B ). (vii) (GL(4) GL(1))/{(z, z 2 ) z GL(1)} GSO(3, 3). (viii) (GL(2, B) GL(1))/{(z, z 2 ) z GL(1)} GSO(5, 1). (iv) (GL(2) GL(2))/{(z, z 1 ) z GL(1)} GSO(2, 2). (x) (B GL(2))/{(z, z 1 ) z GL(1)} GSO (4). 5 Lie Lie ( )dual pair. GL(m, D) GL(m, D), Sp(m, R) O(p, q), U(m, n) U(m, n ), Sp(p, q) O (2n). D. R, C Hamilton H. ( ) ([Mat] ), dual pair. Lie, Accidental dual pair., 18

65 55 Lie, dual pair., SL(2, R) = Sp(1, R).,. 5.1 Lie (1) SL(2, R) O(2, 1) Accidental SL(2, R)/{±1 2 } SO 0 (2, 1)(= O 0 (2, 1)). ( [Shim], [Ni]), ( [Shin-1]). Hecke,, Hecke [Shim]. [Shin-1] Weil, [Ni] Weil. Maass ([K- S], [Ko] ). Waldspurger[W]. [Sak]. (2) SL(2, R) O(2, 3) Accidental Sp(2, R)/{±1 4 } SO 0 (2, 3)(= O 0 (2, 3)). ( [Kur], Andrianov[An], Maass[Ma], Zagier[Za-1]). -. Ramanujan- Petersson, [Kur]. Maass[Maa], Zagier[Za-1] [Ib]. Piatetski-Shapiro[Ps]. 19

66 56 (3) Sp(2, R) O(4) Accidental (Sp(1) Sp(1))/{±(1, 1)} SO(4) ( [Ys]).. Abel Hasse-Weil L- L-. [Ok]. [B-S]. (4) SL(2, R) O(2, 2) Accidental (SL(2, R) SL(2, R))/{±(1 2, 1 2 )} SO 0 (2, 2) Hilbert ( - [D-N]). Dedekind L L, (Base change lift) ([La], [Sat], [Shin- 2] ). [D-N], Accidental ([Kud-1])., Zagier identity - [Za-1]. (5) SL(2, R) O(3, 1) Accidental SL(2, C)/{±1 2 } SO 0 (3, 1)(= O 0 (3, 1)) ( SL(2, C)) ( [As]) -,.,. [Fr]. 20

67 57 (6) U(1, 1) U(1, q) Accidental SU(1, 1) SL(2, R) ( SU(1, q)) (Kudla [Kud-2]). Kudla. q = 2, [Kud- 3], [M-S]. (7) O (4) Sp(1, q) Accidental SO (4) (Sp(1) SL(2, R))/{±(1, 1 2 )} Sp(1, q).,, Kudla Sp(1, q),. ([Na]),. (8) ([Od], [Su]), (1), (2), (4). 3,, - Zagier identity ([Za-1] ) IV. Gan-Takeda[G-Tk] Gan-Tantono[G-Tn] similitude ([G-Tk], [G-Tn] GSp(4) ) Langlands, Accidental (4.4 ). Accidental. 21

68 Lie 5.1. Lie G 1 G 2, G 1 G 2 G 1 G 2, G 1 G 2 G 1 G 2.. ( ) S (1) k+ 1 2 S (1) 2k - S (1) k S (1) k S (1) 2k S(1) k+ 1 2 S (2) k+1 S Hilb (k,k) SL(2, R) O(2, 1)( SL(2, R)) O(2, 1)( SL(2, R)) SL(2, R) SL(2, R) O(3, 2)( Sp(2, R)) SL(2, R) O(2, 2)( SL(2, R) SL(2, R)) S (1) k A 1 2 (k2 1) 2k+1 (SL(2, C)) SL(2, R) O(3, 1)( SL(2, C)) A k1 (B ) A k2 (B ) M (2) (l 1,l 2 ) O(4)( H /R H /R ) Sp(2, R) (even) S (1) k n+1 SIV,2n k SL(2, R) O(2n, 2) (odd) S (1) k n S IV,2n 1 k Kudla S (1) µ ν+1 q M I,q S (1) k SL(2, R) O(2n 1, 2) (µ,ν) SL(2, R) SU(1, 1) SU(1, q) A k (B ) S QDS (0,k) (Sp(1, q)) SO (4)( SL(2, R) H /R ) Sp(1, q) k, k 1, k 2, µ, ν. SL(2, R) SL(2, R). S κ (1) κ. S (1) κ + 1. κ M (2) Siegel ( (l 1, l 2 ) = (k 1 + k 2 + 2, k 1 k 2 + 2), k 1 k 2 ). S (2) Siegel. S(κ,κ) Hilb (κ, κ) ( )Hilbert. A λ k (SL(2, C)) k SU(2), Casimir λ SL(2, C). 22

69 59 A k (B ) k SU(2) {x H N H (x) = 1} B ( B, ). M I,q (µ,ν) (µ, ν) q (I ) ( (µ, ν) [Kud-2, Section 4]. Kudla µ ν > 2q + 1). S IV,m k k m IV ( k > 2n + 2(even ) k > 2n + 1(odd )). S QDS (0,k)(Sp(1, q)) (0, k) Sp(1, q) ( k > 4q + 2, q = 1 k > 4q = 4 ). [An] A. N. Andrianov, Modular descent and the Saito-Kurokawa conjecture, Invent. Math. 53 (1979) [As] T. Asai, On the Doi-Naganuma lifting associated with imaginary quadratic fields, Nagoya Math. J. 71 (1978) [B-S] S. Böcherer and R. Schulze-Pillot, Siegel modular forms and theta series attached to quaternion algebras. Nagoya Math. J. 121 (1991) [D-N] K. Doi and H. Naganuma, On the functional equation of certain Dirichlet series, Invent. Math. 9 (1969/1970) [E-G-M] J. Elstrodt, F. Grunewald and J. Mennicke, Vahlen s Group of Clifford Matrices and Spin Groups, Math. Z. 196 (1987) [F-H] E. Freitag and C. F. Hermann, Some Modular Varieties of Low Dimension, Adv. Math. 152 (2000) [Fr] S. Friedberg, On the imaginary quadratic Doi-Naganuma lifting of modular forms of arbitrary level, Nagoya Math. J. 92 (1983) [G-Tk] W.T. Gan and S. Takeda, The local Langlands conjecture for GSp(4), to appear in Annals of Math. 23

70 60 [G-Tn] W.T. Gan and W. Tantono, The local Langlands conjecture for GSp(4) II, the case of inner forms, preprint. [He] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Volume 34, American Mathematical Society (2001). [Hu] J. Humphreys, Introduction to Lie algebras and representation theory, Graduate text in Math. 9, Springer Verlag, (1972). [Ib], Saito-Kurokawa lifting for level N,. [K-S] S. Katok and P. Sarnak, Heeger points, cycles and Maass forms, Israel J. Math. 84 (1993) [Kn] A. Knapp, Lie groups beyond an introduction, Second edition, Birkhäuser, [Ko] H. Kojima, Shimura correspondence for Maass wave forms of half integral weight, Acta Arith. 69 (1995) [Kr] A. Krieg, Modular forms of half-spaces of quaternions, Lecture Notes in Math. vol.1143, Springer-Verlag, (1985). [Kud-1] S. Kudla, Theta functions and Hilbert modular forms, Nagoya Math. J. 69 (1978) [Kud-2] S. Kudla, On certain arithmetic automorphic forms for SU(1, q), Invent. Math. 52 (1979) [Kud-3] S. Kudla, On certain Euler products for SU(2, 1), Compositio Math. 42 (1980/81) [Kur] N. Kurokawa, Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two, Invent. Math. 49 (1978) [La] R. Langlands, Base change for GL(2), Annals of Mathematics Studies 96, Princeton university press, Princeton, N.J., [Maa] H. Maass, Über eine Spezialschar von Mudulformen zweiten Grades, I, II, III, Invent. Math. 52 (1979) , 53 (1979) , 53 (1979) [Mat], Weil Howe duality,. 24

71 61 [M-S] A. Murase and T. Sugano, On the Fourier-Jacobi expansion of the unitary Kudla lift, Compositio Math. 143 (2007) [Na] H. Narita, Theta lifting from elliptic cusp forms to automorphic forms on Sp(1, q), Math. Z. 259 (2008) [Ni] S. Niwa, Modular forms of half integral weight and the integral of certain theta functions, Nagoya Math. J. 56 (1975) [Od] T. Oda, On modular forms associated with indefinite quadratic forms of signature (2, n 2), Math. Ann. 231 (1977/1978) [Ok] T. Okazaki, Proof of R. Salvati Manni and J. Top s conjectures on Siegel modular forms and abelian surfaces, Amer. J. Math. 128 (2006) [Ps] I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting, Invent. Math. 71 (1983) [Sa] H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in mathematics, vol.8, Kyoto Univ., Kyoto, Japan, (1975). [Sak], Shimura,. [Sat] I. Satake, Classification theory of semi-simple algebraic groups, Lecture Notes in Pure and Applied Mathematics, 3. Marcel Dekker, Inc., New York, [Shim] G. Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973) [Shin-1] T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975) [Shin-2] T. Shintani, On liftings of holomorphic cusp forms, Proc. Sympos. Pure Math. Vol.33, Amer. Math. Soc., Providence, R.I., (1979) part2, [Su], Oda Lift,. [Yk],, (1993). [Ys] H. Yoshida, Siegel s modular forms and the arithmetic of quadratic forms, Invent. Math. 60 (1980)

72 62 [W] J. L. Waldspurger, Correspondence de Shimura, J. Math. Pures et appl., 59 (1980) [Za-1] D. Zagier, Modular forms associated to real quadratic fields, Invent. Math., 30 (1975) [Za-2] D. Zagier, Sur la conjecture de Saito-Kurokawa (d aprés H. Maass), Seminar on Number Theory, Paris , , Progr. Math., 12, Birkhäuser, Boston, Mass.,

73 63 Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo] 2 Hecke ( ) 0 1n J n =, Γ = Γ n = Sp(n, Z) = {γ GL(2n, Z) t γj 1 n 0 n γ = J n } G = GSp(n, Q) + = {g M 2n (R) t gj n g = ν(g)j n, ν(g) R >0 } (2.1) g G g 1 Γ g Γ commensurable [Γ : Γ g 1 Γ g] <, [g 1 Γ g : Γ g 1 Γ g] < Γ G Hecke H(Γ, G) = H(Γ, G) C 1

74 64 C H(Γ, G) = a i (Γ g i Γ ) g i G, a i C i: : L(Γ, G) = { i: a i(γ g i ) g i G, a i C} C- Γ (γ, i a i(γ g i )) i a i(γ g i γ) L(Γ, G) Γ L(Γ, G) Γ ( )( ) a i (Γ γ i ) b j (Γ δ j ) = a i b j (Γ γ i δ j ) j i,j C- i H(Γ, G) L(Γ, G) Γ, Γ gγ = r Γ g i Γ gγ i=1 r Γ g i ( Γ \Γ gγ Γ g 1 Γ g\γ [Γ gγ : Γ ] < ) H(Γ, G) i=1 2.1 H(Γ, G) 2.2 g G Γ gγ diag(a 1,..., a g, d 1,..., d g ) a i d i = ν(g), a i a i+1, a g d g (ν(g) (2.1)) [An1, Theorem 3.28] ( ) ( ) A B t 2.1 G involution g = G g D t B = = C D t C t A ν(g)g 1 α α = J n αjg 1 g G Γ gγ = Γ g Γ Hecke Hecke G (p) = G GL(2n, Z[1/p]) H(Γ, G) = H p, H p = H(Γ, G (p) ) p: g G (p) ν(g) p H p 2

75 ( 1n 0 T (p) = Γ 0 p1 n ), T i (p 2 ) = Γ 1 i p1 n i p 2 1 i p1 n i Γ (0 i n) H p = C[T (p), T i (p 2 ) (1 i n)] (cf. [An1, Theorem 3.40]) Hecke Satake 3 Hecke Satake H p = G p = GSp(n, Q p ), K p = G p GL(2n, Z p ) { ϕ: G p C ϕ K p - i.e. ϕ(k 1 gk 2 ) = ϕ(g), k 1, k 2 K p ϕ } H p convolution ϕ 1 ϕ 2 (h) = ϕ 1 (g)ϕ 2 (g 1 h)dg G p C- 3.1 H p = H(Γ, G (p) ) H p, Γ gγ ch(k p gk p ) C- H p GSp T = {diag(u 1,..., u g, v 1,..., v g ) u 1 v 1 = u 2 v 2 = = u g v g } GSp ( ) 1 A B N = 0 ν t A 1 GSp A =... 1 P = T N GSp minimal parabolic subgroup GSp T Hecke : H p (T ) = {ψ : T (Q p ) C T (Z p )- } 3

76 66 T GL 1 n + 1 H p (T ) = C[X ± 0, X± 1,..., X± n ] ( ( 1n 0 X 0 = ch T (Z p ) 0 p1 n ( ( ) Si 0 X i = ch T (Z p ) 0 S 1 T (Z p ) i ) ) T (Z p ), ) i, S i = diag( 1,..., 1, p, 1,..., 1) T GSp N(T ) = {g GSp g 1 T g T } GSp Weyl W = N(T )/T T 3.2 W S n (Z/2Z) n S n n T t = diag(u 1,..., u n, v 1,..., v n ) n {(u 1, v 1 ), (u 2, v 2 ),..., (u n, v n )} (Z/2Z) n ε i ( i 1 0 ) u i v i W T T Hecke H p (T ) H p (T ) = C[X ± 0, X± 1,..., X± n ] W = S g (Z/2Z) n S g X 1,..., X n (Z/2Z) n ε i X 0 X 0 X i X i X 1 i X j X j (j i) H p (T ) W - H p (T ) W H p H p (T ) C- (Satake ) S : H p H p (T ), S(f)(g) = δ(t) 1/2 f(tn) dn δ(t) modulus n = Lie(N) δ(t) = det(ad n (t) p t = diag(u 1,..., u g, u 0 u 1 1,..., u 0u 1 n ) δ(t) = u n(n+1)/2 0 u 2 1u 4 2 u 2n n N 3.3 Satake C- H p H p (T ) W = C[X 0 ±, X± 1,..., X± n ] W 4

77 67 [Sa] p Bruhat-Tits [Ca] H p M G p KMK KMK = i M i K, M i = p l = ν(m) i ( ) p r i1 0 Ai B i 0 p l t A 1, A i =... i p r in 3.4 M n ch(kmk) X0 l (p j X j ) r ij i j=1 H p C[X ± 0, X± 1,... X± n ] W C X 0 p 3.3 n S(ch(KMK)) p ln(n+1)/4 X0 l (p j X j ) r ij i j=1 (cf. [AS, Lemma 1]) Siege L H n = {Z M n (C) t Z = Z, Im(Z) > 0 ( )} Siegel GSp(n, R) + H n ( ) g Z = (AZ + B)(CZ + D) 1 A B H, g = G C D g GSp(n, R) + k Z H n f f k g(z) = ν(g) nk n(n+1)/2 det(cz + D) k f(g Z ) (4.1) 4.1 H n f Γ = Sp(n, Z) k Siegel 5

78 68 (1) γ Γ f k γ = f (2) n = 1 f cusp Fourier f(z) = ν=0 a(ν)e 2πiνz n 2 (2) (Koecher ) f Fourier f(z) = S n A 0 C(A)e 2πi Tr(AZ) (4.2) S n A 0 A f (3) det(im(z)) k/2 f(z) f cusp k M k (Γ ) cusp S k (Γ ) Siegel Hecke H(Γ, G) Γ gγ Γ γγ = r i=1 Γ γ i f k [Γ gγ ] = r f k γ i H(Γ, G) M k (Γ ) S k (Γ ) i=1 [Gu1] Siegel A = A Q Q GSp(n, A) = GSp(n, Q)GSp(n, R) + p GSp(n, Z p ) ( ) g = γg k f M k (Γ ) GSp(n, A) φ f ( ) φ f (g) = det(ci + D) k A B f(g i1 n ), g = C D Siegel GSp(n, A) g G p f k [Γ gγ ] = φ f ch(k p gk p ) 6

79 69 convolution, GSp(n, A) φ, ψ φ ψ(g) = (n, A)φ(gh)ψ(h 1 ) dh GSp S k (Γ ) Petersson, Z = X + iy H n H n d Z d Z = det Y (g+1) dxdy dx dy Lebesgue d Z Sp(n, R) f, g S k (Γ ) f, g = f(z)g(z) det(y ) k d Z Γ \H n S k (Γ ) cusp Γ αγ H(G) f k Γ αγ, g = f, g k Γ αγ H(G) S k (Γ ) H(G) f S k (Γ ) Hecke X H(G) f X = λ(x) f f 3.4 p λ f : H p = C[X 0 ±, X± 1,..., X± n ] W C C[X 0 ±, X± 1,..., X± n ] C[X 0 ±, X± 1,..., X± n ] W integral λ f C[X 0 ±, X± 1,..., X± n ] C : λ f X 0, X 1,..., X n C- H p C (C ) n+1 /W 4.2 f p (n + 1) {α p,0, α p,1,..., α p,n } f Satake Siegel L Satake Satake Hecke L Weyl 7

80 (1) f {α p,0, α p,1,..., α p,n } Satake 2 L L p st(f, t) = (1 t) 1 n i=1 L p spin (f, t) = (1 α p,0t) 1 ( 1 αp,i t ) 1( 1 αp,i 1 t ) 1 n (1 α p,0 α p,i1 α p,ir t) 1 r=1 1 i 1 < <i r n (2) f s C L st (f, s) = p L spin (f, s) = p L p st(f, p s ) L p spin (f, p s ) Re(s) 0 s L st (f, s) standard L L spin (f, s) spinor L spinor L Euler L spin (t) L spin (f, t) α i X i C[X 0 ±,..., X± n ] W [t] H p [t] ( ) T (p k ) {g G p M 2n (Z p ) ν(g) p k } H p H p H n (t) = T (p k )t k 2 n 2 P n (t) H n (t) = P n (t)l spin (p n(n+1)/2 t) spinor L k=0 L Sp n ( [Yo], connected L-group ) GSp n GSpin(2n + 1, C) = ( C 1 Spin(2n + 1, C) ) /Z, (Z Spin(2n + 1, C) 2 a ( 1, a) ) Spin n 1 {±1} Spin n SO n 1 8

81 71 standard L Spin n+1 SO n+1 SO n+1 standard spinor L spin SO n+1 2n n L L 4.1 (Böcheler [Bö]) f M k (Γ n ) ε n 1 0 ( ) s + ε n Λ(f, s) = (2π) ns π s/2 Γ Γ(s + k j)l st (f, s) 2 j=1 s- Λ(f, s) = Λ(f, s) 4.2 f M k (Γ n ) L spin (f, s) s- s nk n(n + 1) s n = 2 Andrianov ([An1]) 4.3 (Andrianov) Ψ(f, s) = Γ(s)Γ(s k + 2)(2π) 2s L spin (f, s) Ψ(f, s) Ψ(f, 2k 2 s) = ( 1) k Ψ(f, s) n = 3 Asgari-Schmidt L spin ([AS]) 5 L n M = p1 2n = diag(p,..., p) G Satake (3.4) ch(γ MΓ ) = p n(n+1)/2 X0 2 X 1 X n 9

82 72 (4.1) M f M k (Γ ) f k M = p nk n(n+1) f f Satake {α 0,..., α n } α0α 2 1 α n = p nk n(n+1)/2 (5.1) 5.1 ( ) 1 0 n = 1 Γ = SL(2, Z) T p Γ Γ 0 p Γ ( ) 1 0 Γ = Γ 0 p ( ) p 0 Γ = 0 1 ( ) p Γ 0 b p 1 ( ) p b Γ X 0 + X 0 X 1 C[X ± 0, X± 1 ]W f M k (Γ ) T (p)f = λ p f Satake prameter {α 0, α 1 } (5.1) α 0 + α 0 α 1 = λ p, α 2 0α 1 = p k 1 L spin (f, s) = p = p (1 α 0 p s ) 1 (1 α 0 α 1 p s ) 1 (1 λ p p s + p k 1 2s ) spinor L L L(f, s) Spin 3 = SL 2 GSpin(3, C) = GL(2, C) spinor GL(2, C) standard standard L L st (f, s) = p (1 p s ) 1 (1 α 1 p s ) 1 (1 α 1 1 p s ) 1 1 λ p t + p k 1 t 2 = (1 α 0 t)(1 α 0 α 1 t) α 2 0 = p k 1 α 1 1, α 0 (α 0 α 1 ) = p k 1, (α 0 α 1 ) 2 = p k 1 α 1 L st (f, s k + 1) = p (1 α 2 0p s ) 1 (1 α 2 0α 1 p s ) 1 (1 α 2 0α 2 1p s ) 1 = L(f, s, Sym 2 ) = ζ(2s 2k + 2) 10 n=1 a(n 2 ) n s

83 73 ( [Is] ) standard L 2 L accidental Spin(3, C) = SL(2, C) 2:1 SO(3, C) GL(3, C) ([Na2] : SL(2, R) 2:1 SO(2, 1) C ) SL 2 2 L standard L 5.2 Eisenstein Siegel Eisenstein E n k (Z) = ( A B C D ) Γ n \Γ n det(cz + D) k k n + 2 {( ) } Γ n A B = Γ n 0 D k n = 1 2ζ(k)Ek(z) 1 = (cz + d) k (c,d) Z 2 (0,0) Ek n(z) M k(γ n ) Ek n (Z) Hecke ( Zharkovskaya ) 5.1 Siegel Eisensetein E n k Satake α 0 = 1, α i = p k i (1 i n) n = 1 Ek 1 T p 1 + p k 1 n Zharkovskaya Siegel Φ 5.1 f M k (Γ n ) Φ(f) M k (Γ n 1 ) Φ(f)(z) = lim λ f (( )) z 0, z H 0 iλ n 1 11

84 74 f Fourier (4.2) Φ(f) Φ(f)(z) = Fourier (cf [Gu2, 3.3]). S n 1 a 0 C Φ(E n k ) = E n 1 k (( )) a 0 e 2πiaz (Zharkovskaya ) C- ψ : H p (Γ n ) H p (Γ n 1 ) (X 0, X 1,..., X n 1, X n ) (p k n X 0, X 1,..., X n 1, p n k ) f M k (Γ n ) T H p (Γ n ) Φ(f k T ) = Φ(f) k ψ(t ) [An2, Theorem 4.19] n = 2 Zharkovskaya Satake α 0 = p k 2, α 1 = p k 1, α 2 = p 2 k 3.2 W ε 2 α 0 = 1, α 1 = p k 1, α 2 = p k n L st (Ek n, s) = ζ(s) ζ(s k + i)ζ(s + k i) i=1 [An1] A.N. Andrianov Euler prodcts associated with Siegel modular forms of degree two, Russ. Math. Surveys 29, 3, (1974). [An2] A.N. Andrianov Indroduction to Siegel modura forms and Dirichlet series, Universitext. Springer, New York (2009). [AS] M. Asgari and R. Schmidt, Siegel modular forms and representations. Manuscripta Math. 104, no. 2, (2001). 12

85 75 [Bö] S. Böcherer Über die Funktionalgelichung automprpher L-Funktionen zur Siegelschen Modulgruppe. J. reine angew. Math. 362, (1985). [Ca] P. Cartier, Representations of p-adic groups: a survey. Automorphic forms, representations and L-functions, Part 1, pp , Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence, R.I. (1979). [Fr] E. Freitag, Siegelsche Modulfunktionen, Grundl. Math. Wiss Springer-Verlag, Berlin. [vg] G. van der Geer, Siegel modular forms and their applications. The of modular forms, , Universitext, Springer, Berlin, (2008) [Gu1], GL 2, [Gu2], Siegel Eisensetein Fourier, [Is], L, 16 L (2009), p3-36. [Na1], L, 16 L (2009), p [Na2], Accidental, [Sa] I. Satake, Theory of spherical functions on reductive groups over p-adic fields, Publ. Math. I.H.E.S. 18, 5-69 (1963). [Yo], Functoriality Principle, 13

86 76 Weil Howe duality 2011 Weil Howe duality Weil Howe duality 1 Weil 1.1 Fourier F. T 1 F unitary character χ F : F T χ F cannical F χ F 1. χ R (t) = e 2πit (t R). 2. F p- Q p χ Qp χ Qp (Z p ) = 1 k Z p v χ Qp (vp k ) = e 2πivpk 3. F Q p χ F = χ Qp Tr F/Qp dx F Haar F Borel (Haar 1

87 77 F = R dx Euclid C 0 (F ) = {f : F C f f } L 2 (F ) = { f : F C f F f(x) 2 dx < } L 2 (F ) F f(x) 2 dx 2 Hilbert Fourier f C 0 (F ) F(f)(a) = χ F (ax) 1 f(x)dx. F Hilbert F : L 2 (F ) L 2 (F ) Haar dx F 2 = F F : L 2 (F ) L 2 (F ) F 2 (f)(x) = f( x) f L 2 (F ) Planchrel dx Haar F = R Euclid F Q p χ F F (Nd) 1/2 Haar Nd F absolute differential F Haar Fourier V F - n V V, V V F canonical pairing V v 1,..., v n V Haar ( n ) f(v)dv = f x k v k dx 1 dx n V F F 2 L 2 (V ) 1 Fourier F : L 2 (V ) L 2 (V ) Ff(ξ) = χ F ( v, ξ ) 1 f(v)dv V v 1,..., v n V v 1,..., v n V V F : L 2 (V ) L 2 (V ) F 2 f(v) = f( v) f L 2 (V ) Fourier 2 k=1

88 Heisenberg Stone-von Neuman X F - dim(x) = 2n X, x, x = 0 (x X) (X,, ) 1. X V V = {x X x, v = 0 (v V )} X V V = V Lagrangean L(X) X Lagranean 2. X Sp(X) Sp(X) = {g GL(X) gx, gy = x, y (x, y X)}. 3. X v 1,..., v n, w 1,..., w n v i, v j = 0 (1 i, j n), w i, w j = 0 (1 i, j n), v i, w j = δ i,j (1 i, j n). δ i,j Kronecker V L(X) dim V = n. 2. Sp(X) L(X) 3. V, V L(X) transversal V V perfect pairing V V V v 1,..., v n V w 1,..., w n V v 1,..., v n, w 1,..., w n X Heisenberg V, V L(X) transversal X = V V Heisenberg H(V, V ) H(V, V ) = V V F (v, w, a)(v, w, b) = (v +v, w+w, v, w +a+b) (v, v V, w, w V, a, b F ) H(V, V ) 3

89 (Stone-von Neuman V, V L(X) transversal v 1,..., v n V V Haar dv 2 L 2 (V ) U(L 2 (V )) L 2 (V ) T : H(V, V ) U(L 2 (V )) T ((v, w, a)φ(x) = χ F ( x, w + a)φ(x + v) (v V, w V, a F, x V ) H(V, V ) L 2 (V ) Stonevon Neuman (Stone-von Neuman 1. Stone-von Neuman (T, L 2 (V )) {0} 2. H(V, V ) (π, H) π((0, 0, a))h = χ F (a)h (a F, h H) Hilbert isometry) J : L 2 (V ) H J T ((v, w, a)) = π((v, w, a)) J (v V, w V, a F ) J 1 F Sp(X) Heisenberg Weil Heisenberg Sp(X) Heisenberg Heisenberg Heisenberg F H(X) = X F (x, a)(x, b) = (x + x, 1 2 x, x + a + b) (x, x X, a, b F ). I : Sp(X) Aut(H(X)) I(g)((x, a)) = (g 1 x, a) (g Sp(X), x X, a F ) V, V L(X) transversal Φ : H(V, V ) H(X) Φ(v, w, a) = (v + w, 1 v, w + a) 2 H(X) Stone-von Neuman T : H(X) U(L 2 (V )) Φ Φ((0, 0, a)) = (0, a) (a F ) 4

90 Weil (F, χ F, dx) (X,,, ) F transversal V, V L(X) V v 1,..., v n V Haar dv 2 L 2 (V ) g Sp(X) g Stone-von Neuman ( T, L 2 (V )) T g = T I(g) ( Mp(X) Mp(X) = {(g, J) Sp(X) U(L 2 (V )) J 1 T ((x, a)) J = T g ((x, a)) x X, a F }. Mp(X) Sp(X) U(L 2 (V )) ι : T Mp(X) ι(a) = (id X, aid L 2 (V )) (a T) p 1 : Mp(X) Sp(X) Sp(X) U(L 2 (V )) Sp(X) Mp(X) T ι Mp(X) p 1 Sp(X) 1 Ker(p 1 ) = Im(ι) Schur p 1 T g ((o, a)) = χ F (a)id L 2 (V ) Stone-von Neuman (Weil W : Mp(X) U(L 2 (V )) 2 Sp(X) U(L 2 (V )) U(L 2 (V )) Mp(X) (W, L 2 (V )) Weil Weil Sp(X) Weil Sp(X) double cover ([Weil]) Mp(X) Sp(X) Im(ι) Sp(X) = {ι(1), ι( 1)} p 1 ( Sp(X)) = Sp(X) 5

91 81 V v 1,..., v n V w 1,..., w n v 1,..., v n, w 1,.., w n Sp(X) v 1,..., v n ( w 1,..,) w n Sp(X) α β block-wise α(resp. γ) F γ δ n n End F (V ) (resp. Hom F (V, V )) ( ) 0 I n σ =, I n 0 {( ) } α 0 L = 0 t α 1 α GL(V ), {( ) } N = I n 0 γ I n γ F n n σ Sp(X) L N Sp(X) Sp(X) L,N σ Weil L,N σ Sp(X) (Weil constant cf. [Ikeda]) a F γ χf (a) T F Schwartz φ ( ) ax 2 ( ) φ(x)χ F dx = γ χf (a) a 1 2 F Fφ(x)χ F x2 dx 2 2a F γ χf (a) Weil constant ( ) α 0 1. g = L 0 t α 1 W (g) U(L 2 (V )) 2. g = ( W (g)φ(v) = ) I n 0 γ I n F γ χf (1) 1 2 det γ χf (det α 1 a F ) φ(α 1 v) (φ L 2 (V )) N W (g) U(L 2 (V )) W (g)φ(v) = χ F ( 1 2 v, γv ) φ(v) (φ L 2 (V )). 3. W (σ) U(L 2 (V )) W (σ)φ = γ F (1) n Fφ (φ L 2 (V )) g L N {σ} (g, W (g)) Sp(X) 6

92 82 2 Howe duality F = R X 2n, 2.1 Howe duality G H Z G (H) = {g G hg = gh(h H)} H 1, H 2 Sp(X) reductive (H 1, H 2 ) reductive dual pair Z Sp(X) (H 1 ) = H 2 Z Sp(X) (H 2 ) = H p,q,m n = 2(p + q)m n X = R p+q R 2m (, ) p,q R p+q (p, q), m R 2m, = (, ) p,q, m X O(p, q) = {g GL(p + q, R) (gv, gw) p,q = (v, w) p,q (v, w R p+q )} O(p, q) Sp(m.R) Sp(X) (O(p, q), Sp(m.R)) reductive dual pair Howe duality (H 1, H 2 ) reductive dual pair H 1 H 2 H 1 H 2 Sp(X) Weil (W, L 2 (V )) Sp(X) H 1 H2 H 1 L 2 (V ) = ˆ i=0 σ i τ i σ i τ i H 1 H 2 Howe duality σ i (i = 0, 1, 2,...) τ i (i = 0, 1, 2,...) Weil H 1 H 2 Howe Theta lifting H 1 7

93 83 Howe duality Weil reductive (Harish-Chandra 2.2 Harish-Chandra G reductive K g 0 k 0 G K g k U(g) g K K V K (π, V ) K- v V v K- K V U(k)- 2. K- V δ K V δ-isotypical component V (δ) V (δ) = Im(φ) φ Hom K (δ,v ) V = δ K V (δ) 3. K- V admissible δ K dim V (δ) < 4. V (g, K)- (a) V U(g)- K- (b) k K, X g, v V k X k 1 v = (Ad(k)X) v (c) K- U(k)- U(g)- U(k)- 5. (g, K)- Harish-Chandra Harsh-Chandra admissible Harish-Chandra G (π, H) G Hilbert 8

94 84 1. H K-finite part H K H K- H H K K- 2. H K admissible (π, H) admissible admissible Hilbert (π, H) v H K X g 0 π(exp(tx))v t = 0 Xv = d dt π(exp(tx))v t=0 Xv H K H K admissible (g, K) admissible 2. admissible Hilbert (π, H) (g, K)- H K 3. Harish-Chandra V admissible Hilbert (π, H) (g, K)- V H K admissible Hilbert admissible Banach ( Paley-Wiener admissible 2. Harish-Chandra V V = H K admissible Hilbert (π, H) 2.3 Fock model transversal V, V L(X) V v 1,..., v n V Haar dv 2 L 2 (V ) V C V 1.3 V v 1,..., v n V w 1,..., w n ( ) v 1,..., v n, w 1,.., w n Sp(X) 0 I n σ = σ imaginary unit 1 I n 0 X v 1,..., v n X X V C 9

95 85 H(, ) (, ) v 1,..., v n V C Hermite K = {g GL(V C ) H(gv, gw) = H(v, w) (v, w V C )} H(, ) X, K Sp(X) K Sp(X) K K Sp(X) Weil L 2 (V ) (Schödinger model) L Sp(X) K K-finite part Weil Fock model (z 1,..., z n ) v 1,..., v n V C dz = dz 1... dz n, d z = d z 1... d z n dz d z V C Haar (Fock space) O(V C ) V C Fock F { } F = f O(V C ) f(z) 2 e πh(z,z) dz d z < V C F V C f(z) 2 e πh(z,z) dz d z 2 Hilbert (Bargman cf. [F]) B : L 2 (V ) F Bφ(z) = V φ(x)e π(2(x,z) (x,x) 1/2(z,z) dx L 2 (V ) F isometry F Weil Weil Fock model P V C C- P = C[z 1,..., z n ]. P F k k P k = {f P f(tv) = t k f(v) (t C, v V C )} Weil (W, F) admissible K-finite part F K P P K- P = P k k=0 10

96 86 K K k g Sp(X) Sp(X) q g k Killing form ( )Cartan g = k q q K K q +,q q = q + q ω : U(g) End C (P) Weil U(g)- z i (1 i n) { ( ) } 1 1. ω(k) z 2 i z j + z i z j 1 i, j n { 2. ω(q + ) 2 z i z j 1 i, j n } 3. ω(q ) {z i z j 1 i, j n} k ω(q + )P k = P k 2, ω(q )P k = P k+2 P even = k=0 P2k P odd = k=0 P2k+1 P = P even P odd P 2.4 ([H1],[KV]) (H, H ) Sp(X) reductive dual pair H h h H H H K H K H h ± = h q ±, r = h k, R = H K H R H R Sp(X) K P H- R- isotypical component σ H σ-isotypical component P(σ) H- (h, R) K S R(S; P) = {σ S P(σ) 0}. 2. H(H) H(H) = {f P ω(y )f = 0 (Y h +)}. 11

97 87 3. σ R( H; P) deg(σ) = min{k P k P(σ) 0} H(H) H R σ R( H; P) 1. H(H)(σ) = P(σ) P deg(σ). H(H)(σ) 0 2. P(σ) = U(h )H(H)(σ). 3. H(H)(σ) H R- 4. ψ : R( H; P) R( R; P) H R- H(H)(σ) = σ ψ(σ) H(H) H(H) = σ ψ(σ). σ R( H;P) τ R Verma M(τ) M(τ) = U(h ) U(r+h + ) τ τ h + 0 M(τ) Harish-Chandra (h, R)- 2. M(τ) (h, R)- L(τ) ([H1]) ψ : R( H; P) R( R; P) σ R( H; P) P(σ) = σ L(ψ(σ)) H (h, R)- P = σ L(ψ(σ)), σ R( H;P) 12

98 ([H2]) (H, H ) Sp(X) reductive dual pair H h h H H R = H K R = H K H H R(h, R; P) Harish-Chandra(h, R)- P R(h, R ; P) 2. ρ R(h, R; P) N ρ P/N = ρ P (h, R)- N N ρ = N. P/N =ρ ([H2]) N ρ P h h, R R )- (h, R )- θ(ρ) P/N ρ = ρ θ(ρ) ([H2]) ρ R(h, R; P) (h, R )- θ(ρ) quasi-simple Harish-Chandra θ(ρ) θ(ρ) R(h, R ; P) H H (Howe duality) θ : R(h, R; P) R(h, R ; P) [H2] Howe duality Howe duality [H2] Howe duality ρ R(h, R; P) σ R ρ R R degree σ R( R ; P) 13

99 89 1. σ θ(ρ) θ(ρ) R degree 2. deg(σ) = deg(σ ). 3. Z ρ = P/N ρ = ρ θ(ρ) σ σ Z ρ R R - H σ,σ Z(σ) = U(h )H σ,σ, Z(σ ) = U(h)H σ,σ σ σ Joint harmonics Case-by-case analysis Howe M = Z Sp(X) (R) M = Z Sp(X) (R ) (R, M), (R, M ) reductive dual pair R R (R, M), (R, M ) H( R) H( R ) H( R) H( R ) joint harmonics R( R; P) = {σ R( R; P) H( R)(σ) H( R ) 0}, R( R ; P) = {τ R( R ; P) H( R) H( R )(τ) 0}, joint harmonics ([H2]) ξ : R( R; P) R( R ; P) 1. σ R( R; P) deg(σ) = deg(ξ(σ)) 2. σ R( R; P) H( R)(σ) H( R ) = H( R) H( R )(ξ(σ)) 3. σ R( R; P) R R - H( R)(σ) H( R ) = σ ξ(σ) 14

100 90 4. joint harmonics R R - H( R) H( R ) = H( R)(σ) H( R ). σ R( R;P) 5. ρ R(h, R; P) σ R ρ R R degree σ ξ(σ) joint harmonics Howe 3 [S] 4 Weil Weil Howe duality Fourie [T] Tate J. T. Fouriern analysis in number fields and Hecke s zeta-functions in Cassels and Fröhlich (ed. ) Algebraic Number Theory Academic Press Fock model Bargman Satake [F] Folland G. B. Harmoinc analysis in phase space, Annales of Mathematcs Studies, 122 Princeton University Press, Princeton NJ 1989 Weil Weil [W] Weil, A. Sur certains groupes d operateurs unitaires, Acta. Math. 111,(19649, [Ku] Kudla, S. S. Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87(1994), [R] Ranga Rao,R. On some explicit formulqas in the theory of Weil representation, PacificJ. Math. 157(1993), [LV] Lion, G. A., Vergne, M. The Weil representation, Maslov index and theta series, Progress in Mathematics, 6 Birkhäusr, Boston, MA Weil constant [S] reductive dual pair [S] reductive dual pair Howe [H1] Howe, R. Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), [KV] Kashiwara, M,, Vergne, M. On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44(1978),

101 91 [S] Howe duality [H2] Howe, R. Transcending classical invariant theory, J. Amer. Math. SOc. 2(1989), Howe duality [A] Adams, J. D. Discrete spectrum of the reductive dual pair (O(p, q), Sp(2m)). Invent. Math. 74 (1983), [M] Moeglin, C. Correspondance de Howe pour les paires reductives duales: quelques calculs dans le cas archimedien, J. Funct. Anal. 85 (1989),

102 92 Siegel Jacobi I G K- δ Hecke K- δ Siegel Harish-Chandra Siegel...25 II Siegel 28 4 Weil Heisenberg Weil Kashiwara-Maslov Schrödinegr Weil Jacobi

103 93 5 1/ Fock Siegel Howe III Jacobi Jacobi Harish-Chandra GSp(V ) Jacobi Jacobi Jacobi Siegel Eichler-Zagier, Ibukiyama revisited Jacobi Hecke Weil Jacobi σ = [ a c ] b Sp(n, R) (z,w) H n C n d σ(z, w) =(σ(z), wj(σ, z) 1 ) σ(z) =(az + b)(cz + d) 1, J(σ, z) =cz + d 0 <k Z H n C n F 2

104 94 1) x, y Z n F (z,w+xz+y) =e( ( x, xz +2 x, w ))F (z,w), e(t) =exp2π 1t x, y = x t y 2) σ Sp(n, Z) F [σ] k = F (F [σ] k )(z, w) =F (σ(z),wj(σ, z) 1 )detj(σ, z) k e( w t c, wj(σ, z) 1 ) [ ] a b (σ = ) c d F Fourier F (z,w) = N Sym n (Z),r Zn a(n,r)e(tr(nz)+ r, w ). Sym n(z) ={N Sym n (Q) tr(ns) Z for S Sym n (Z)}. F 1), 2) 3) a(n,r) 0 N 4 1 t rr 0 F k, index 1 Jacobi J k,1 F J k,1 3) 3) a(n,r) 0 N 4 1 r t r>0 F Jacobi J cusp k,1 F J k,1 1) F w C n Riemann ϑ m,m (z,w) = ( ) 1 e 2 p + m /2, (p + m /2)z + p + m /2,w+ m /2 p Z n (z H n,w C n ) θ μ (z,w) =ϑ μ,0 (2z,2w) F (z,w) = σ(f )(z) = μ Z n /2Z n F μ (z)θ μ (z,w) θ(z) =ϑ 0,0 (2z,0) = μ Z n /2Z n F μ (4z) (z H n ) p Z n e( p, pz ) (z H n ) 3

105 95 H n h σ Sp(n, R) (h [σ] k 1/2 )(z) =h(σ(z)) θ(σ(z)) θ(z) Γ 0 (4) = σ = {[ a [ a c c det J(σ, z) k ] } b Sp(n, Z) d c 0 (mod 4) ] b Γ 0 (4) d ( ) 2 θ(σ(z)) = sign(det c)detj(σ, z) θ(z) H n h 1) σ Γ 0 (4) h [γ] k 1/2 = h, 2) h Fourier h(z) = c(t )e(tr(tz) 0 T Sym n (Z) h Γ 0 (4) k 1/2 Siegel M k 1/2 (Γ 0 (4)) 2) 2) c(t ) 0 T>0 h S k 1/2 (Γ 0 (4)) h M k 1/2 (Γ 0 (4)) c(t ) 0 μ Z n s.t. T t μμ (mod 4Sym n(z)) M + k 1/2 (Γ 0(4)) S + k 1/2 (Γ 0(4)) = S k 1/2 (Γ 0 (4)) M + k 1/2 (Γ 0(4)) Eichler-Zagier [3] (n =1 ) Ibukiyama [10] (n 1 ) F σ(f ) J k,1 M + k 1/2 (Γ 0(4)) J cusp k,1 S + k 1/2 (Γ 0(4)) Hecke 4

106 96 I Γ D V J :Γ D GL C (V ) J(γγ,z)=J(γ,γ z) J(γ,z) (γ,γ Γ,z D) (1) J γ Γ F (γz) =J(γ,z)F (z) (z D) F : D V Γ D (1) Γ D V γ(z,v) =(γz,j(γ,z)v) X =Γ\D, V =Γ\(D V ) π : V X ((z,v) (mod Γ) ż = z (mod Γ)) z D Γ z = {γ Γ γz = z} J(γ,z) =1 V (γ Γ z ) v (z,v) (modγ) V π 1 (ż) π : V X X F : D V Γ D ż (z,f(z)) (mod Γ) π : V X [1] 1.2 G D Γ G J G J : G D GL C (V ) J(gg,z)=J(g, g z) J(g,z) (g, g G, z D) (2) o D K = {g G go = o} K V δ δ(k) =J(k, o) Γ F : D V f F : G V f F (g) =J(g, o) 1 F (go) (g G) 1) γ Γ f F (γg)=f F (g), 2) k K f F (gk) =δ(k) 1 f F (g) F (z) =J(g, o)f F (g) (z = go D,g G) D G 5

107 97 V V v V,α V v, α = α(v) C K (δ, V ) (ˇδ, V ) v, ˇδ(k)α = δ(k 1 )v, α α V f F,α (g) = f F (g),α (g G) k K f F,α (gk) = f F (g), ˇδ(k)α = f F,ˇδ(k)α (g) (g G) G Γ- L(Γ\G) G ρ r T : V L(Γ\G) (α f F,α ) k K T ˇδ(k) =ρ r (k) T (δ, V ) K f F,α K L(Γ\G) ˇδ G K G G Z(G) A G Γ A Γ/A Γ G, K, A Haar d G (x),d K (k),d A (a) d K (k) =1 G/A Haar d G/A (ẋ) K G ϕ(x)d G (x) = G/A d G/A (ẋ) d A (a)ϕ(xa) A (ϕ C c (G)) 1 Γ Haar d Γ (γ) ϕ(γ)d Γ (γ) = ϕ(γa)d A (a) (ϕ C c (Γ)) Γ γ Γ/A Γ\G G- d Γ\G (ẋ) ϕ(x)d G (x) = d Γ\G (ẋ) d Γ (γ)ϕ(γx) (ϕ C c (G)) G Γ\G ϕ(ẋ)d G/A (ẋ) = G/A A Γ\G γ Γ/A Γ ϕ( γẋ)d Γ\G (ẋ) (ϕ C c (G/A)) 1 C c (G) G 6

108 A χ K (δ, V δ ) e δ (k) =dimδ tr δ(k) (k K) G f 1) a A f(ax) =χ(a) 1 f(x), 2) f(x) d G/A (ẋ) < G/A f(x) d G/A (ẋ) =0 f G/A L 1 (G/A, χ) f = f(x) d G/A (ẋ) G/A Banach (f g)(x) = f(xy)g(y 1 )d G/A (ẏ) G/A f f (f (x) =f(x 1 )) Banach L 1 (G/A, χ) f G G/A f (ẋ) = f(x) C c (G/A, χ) L 1 (G/A, χ) L 1 (G) Banach C c (G) L 1 (G) f C c (G) f χ (x) = f(ax)χ(a)d A (a) (x G) C c (G/A, χ) f χ f f f χ C c (G) C c (G/A, χ) L 1 (G) L 1 (G/A, χ)(f f χ ) G (σ, E) σ A = χ f L 1 (G/A, χ) u, v E (σ(f)u, v) = f(x)(σ(x)u, v)d G/A (ẋ) G/A E σ(f) f σ(f) Banach L 1 (G/A, χ) e δ f = f e δ = f f L 1 (G/A, χ) L 1 (G/A, χ; δ) L 1 (G/A, χ) (e δ f)(x) = e δ (k)f(k 1 x)d K (k), K (f e δ )(x) = f(xk 1 )e δ (k)d K (k) (x G) K k K f(kxk 1 )=f(x) f L 1 (G/A, χ; δ) L 1 (G/A, χ; δ) o L 1 (G/A, χ; δ) C c (G/A, χ; δ) =L 1 (G/A, χ; δ) C c (G/A, χ), C c (G/A, χ; δ) o = L 1 (G/A, χ; δ) o C c (G/A, χ) A 7

109 99 L 1 (G/A, χ; δ), L 1 (G/A, χ; δ) o C c (G/A, χ; δ) o K- δ Hecke f C c (G) e δ f = f e δ = f C(G; δ) k K f(kxk 1 )=f(x) f C c (G, δ) C(G; δ) o f f χ C c (G, δ),c c (G, δ) o C c (G/A, χ; δ),c c (G/A, χ; δ) o C L 1 (G/A, χ; δ) L 1 (G/A, χ; δ) o 1) π A = χ G π π K δ 1 2) L 1 (G/A, χ; δ) o L 1 (G/A, χ; δ) 3) L 1 (G/A, χ; δ) o [8, pp.12 15], [2, 3.6.2] 1 C- L r a i L (1 i r) [a 1,a 2,,a r ]= sign(σ)a σ(1) a σ(2) a σ(r) =0 σ S r L r- 1) L r- s>r L s- 2) dim C L = n< L n +1-3) n C- M n (C) r- r r(n) r(n) n 4) r(n +1) r(n) > <n Z 1) π A = χ G π π K δ n 2) L 1 (G/A, χ; δ) n- 2.3 G π K δ ( 0) π -δ δ π (π, H π ) δ r G π A = χ H π (δ) δ- H π π(e δ ) (π(e δ )u, v) = e δ (k)(π(k)u, v)d K (k) (u, v H π ) K 8

110 100 u π(e δ )u H π H π (δ) Φ π,δ : G End C (H π (δ)), ψ π,δ : G C Φ π,δ (x) =π(e δ ) π(x) Hπ (δ), ψ π,δ (x) =(dimδ) 1 tr Φ π,δ (x) (x G) Φ π,δ (π, H π ) K- δ ψ π,δ π K- δ Φ π,δ (kxk )=π(k) Φ π,δ (x) π(k ) (k, k K) k K ψ π,δ (kxk 1 )=ψ πdelta (x) e δ ψ π,δ = ψ π,δ e δ = ψ π,δ f L 1 (G/a, χ; δ) o π(f)h π (δ) H π (δ) π(f) Hπ (δ) = G/A f(x)φ π,δ (x)d G/A (ẋ) End K (H π (δ)) δ π r H π (δ) =Vδ r End K (H π (δ)) = M r (C) π(f) Hπ (δ) End K (H π (δ)) = M r (C) Φπ,δ (f) Ψ π,δ : L 1 (G/A, χ; δ) o M r (C) C- f L 1 (G/A, χ; δ) o ψ π,δ (f) = f(x)ψ π,δ (x)d G/A (ẋ) G/A ψπ,δ (f) =tr Ψ π,δ (f) ψ π,δ G {g i } i=1,,n G (ψ π,δ (g i g 1 j )) i,j=1,,n Hermite (π, H π ) Φ π,δ : C c (G/A, χ; δ) o M r (C) G ψ 1) a A ψ(ax = χ(a)ψ(x), 2) k K ψ(kxk 1 )=ψ(x), 3) e δ ψ = ψ e δ = ψ, 9

111 101 4) C- Φ:L 1 (G/A, χ; δ) o M r (C) Φ(f )=Φ(f), f(x)ψ(x)d G/A (ẋ) =trφ(f) G/A f L 1 (G/A, χ; δ) o δ r ψ = ψ π,δ G π G G π, π -δ π A = π A = χ π, π ψπ,δ = ψ π,δ 2.4 (π, H π ) G δ 1 π A = χ π G (ρ, V ρ ) Γ π r,ρ =Ind G Γ ρ L 2 (Γ\G, ρ) 1) γ Γ f(γx)=ρ(γ)f(x), 2) f(x) 2 d Γ\G (ẋ) < Γ\G f : G V ρ 0 f (f,g) = (f(x),g(x)) ρ d Γ\G (ẋ) Γ\G Γ\G f(x) 2 d Γ\G (ẋ) = ( Hilbert V ρ (, ) ρ Hilbert x G f L 2 (Γ\G, ρ) (π r,ρ (x)f)(y) =f(yx) L 2 (Γ\G, ρ) f G M G f(x) 2 d G (x) < M π G Ind G Γ ˇρ ˇρ ρ ˇπ- ρ A = χ Ind G Γ ˇρ ˇπ- K ˇδ- 1.2 L 2 (Γ\G, ˇρ;ˇπ, ˇδ) Godemenet 10

112 σ A = χ G (σ, E) E π- E(π) K (σ K,E(π)) δ- E(π, δ) E(π, δ) ={u E σ(f)u = ψ π,δ (f)u for f C c (G/A, χ; δ) o } Ind G Γ ˇρ ˇπ π G f : G Hom C (V ρ,v δ ) 1) γ Γ f(γx)=f(x) ρ(γ) 1, 2) f(x) 2 d Γ\G (ẋ) <, Γ\G 3) k K f(xk) =δ(k) 1 f(x), 4) ϕ C c (G/A, χ; δ) o f(xy 1 )ϕ(y)d G/A (ẏ) = ψ π,δ (ϕ) f(x) G/A A δ (Γ\G, ρ, π) A, B Hom C (V ρ,v δ ) (A, B) =tr(a B ) B Hom C (V δ,v ρ ) (B v, u) ρ =(v, Bu) δ (v V δ,u V ρ ) A δ (Γ\G, ρ, π) (f,g) = (f(x),g(x))d Γ\G (ẋ) G/A f α A δ (Γ\G, ρ, δ) C V δ θ f α : G V ρ θ f α (x),u =(dimδ) 1/2 α, f(x)u (x G, u V ρ ) f α θ f α Hilbert A δ (Γ\G, ρ, π) C V δ L 2 (Γ\G, ˇρ;ˇπ, ˇδ) dim χ =1 V ρ = C A Hom C (V ρ,v δ ) A(1) V δ A δ (Γ\G.ρ.π) 1) γ Γ f(γx)=ρ(γ) 1 f(x), 2) f(x) 2 δd Γ\G (ẋ) <, Γ\G 11

113 103 3) k K f(xk) =δ(k) 1 f(x), 4) ϕ C c (G/A, χ; δ) o f(xy 1 )ϕ(y)d G/A (ẏ) = ψ π,δ (ϕ) f(x) G/A f : G V δ (f,g) = (f(x),g(x)) δ d Γ\G (ẋ) Γ\G Hilbert 2.5 P p P G p K p G p P P p P f = P \ P K p G p p P x p K p (x p ) p P G p p P G G p p P Tikhonov P S P G S = G p K p p S p P\S P S S P G S G S G = G S V G P S P P S P V G S G S G G {K p } p P {G p } p K = K p G p P p P A p Z(G p ) {A p K p } p P {A p } p P A A Z(G) G p Haar μ p p P f μ p (K p )=1 K p (p P) Haar ν p ν p (K p )=1 P S P ν S (K S )=1 K S = K p Haar p S ν S G S Haar μ S = μ p ν S supp ϕ G S p S ϕ C c (G) C c,s (G) C c (G) = C c,s (G) G Haar μ ϕ(x)dμ(x) = ϕ(x)dμ S (x) G G S P S P (ϕ C c,s (G)) 12

114 104 μ Haar {μ p } p P {K p } p P G/A (x p ) p P (mod A) (x p (mod A p )) p P {G p /A p } p P {K p A p /A p } p P p P Gp /A p p P A p Haar η p p P η p (A p K p )=1 A Haar η {η p } p P {A p K p } p P G p /A p, G/A Haar μ p, μ ψ(x)dμ p (x) = G p ϕ(x)dμ(x) = G G p /A p G/A ( ) ψ(xa)dη p (a) dμ p (ẋ) (ψ C c (G p ), A p ( ) ϕ(xa)dη(a) dμ(ẋ) (ϕ C c (G)) A μ {μ p } p P {K p A p /A p } p P K p (p P) K p K p (δ p ) p P p P p P δ p = 1 Kp Kp (δ p ) p P Kp p P p P S P p S δ p = 1 Kp p S K p 1 1 S p P δ p =( p S δ p ) 1 S K (δ p ) p P p P δ p p P Kp K K p (δ p ) p P p P δ p Kp K p P A p P\P A p K p A p A p Pontryagin  p L p =(A p K p ) = {α Âp A p K p,α =1} {Âp} p P {L p } p P  p (α p ) p P  p p P α p  p P p P (a p ) p P α p (a p ) p P 13

115 ) (α p ) p P p P α p p P  p  2) p P A p (δ p ) p P Kp δ = p P δ p K (χ p ) p P  p p P p P χ = p P χ p  C p = C c (G p /A p,χ p,δ p ), H p = C c (G p /A p,χ p,δ p ) o p P f K p G p G p K p ε p C c (G p ) ε χ,p (x) = ε p (xa)χ p (a)dη p (a) (x G p ) A p ε χ,p C p P δ p 1 Kp χ p L p p P f S 1 S 1 p S 1 ε χ,p (1) = 1 ϕ C p ε χ,p ϕ = ϕ ε χ,p = ϕ 1 : ẋ K p A p /A p, ε χ,p (x) = 0 : ẋ K p A p /A p S 1 S P p S ϕ p p S C p G ( p S ϕ p )(x) = ϕ p (x p ) ε χ,p (x p ) (x =(x p ) p P G) p S p S p S C p L 1 (G/A, χ, δ) C- p P C p = S p S C p L 1 (G/A, χ, δ) S S 1 S P L 1 (G/A, χ, δ) o C- p P H p = p S H p S L 1 - p P C p L 1 (G/A, χ, δ) p P H p L 1 (G/A, χ, δ) o Hilbert {H p } p P p P S 0 P u o p =1 u o p H p S P p S H p ( p S u p, p S v p ) = p S(u p,v p ) 14

116 106 S 0 S T P u u ( p T \S u o p) p S H p p T H p p S H p p T H p p P H p = S 0 S P p S H p p P H p p P H p p P u p p P u p = u o p ( p P u p, p P v p )= p P(u p,v p ) p P H p p PH p {H p } p P {u o p} p P\S0 Aut(H p ) (T p ) p P p P p P T p u o p = u o p p P Aut(Hp ) (T p ) p P p P Aut(Hp ) S 0 S P p S T p Aut( p S H p ) S 0 S T P p S H p p T T p = p S T p p P T p Aut( p P H p ) ( p P T p ) p P u p = p P T p u p p P T p Aut( p PH p ) (T p ) p P p P T p Aut(Hp ) Aut( p PH p ) p P p P G p (π p,h p ) p P S 0 P π p Kp K p 1 1 Kp 1 p P \ S 0 u o p =1 K p - u o p H p {H p } p P {u o p} p P\S0 H x =(x p ) p P G p P x p K p (π p (x p )) p P p P Aut(Hp ) π(x) = p P π p (x p ) Aut(H) (π, H) G K p - u o p H p 1 (π, H) {π p } p P {K p } p P p P π p π p p P K p p P π p G p P π p Ap = χ p π A = χ G p (p P) I p P C c (G p /A p,χ p, 1 Kp ) π A = χ G 15

117 107 (π, H) δ = p P δ p K ((δ p ) p P p P Kp ) K- G p π p π = p P π p G π G p π p K p δ = p P δ p K, (δ p ) p P p P Kp K- 1 π p Ap = χ p Âp (χ p ) p P p P Â p π A = χ = p P χ p π p K p - δ p ψ p x =(x p ) p P G p P x p K p δ p = 1 Kp ψ p (x p )=1 ψ π,δ (x) = p P ψ p (x p )(x =(x p ) p P G) G Γ A Γ (ρ, V ρ ) ρ A = χ G A δ (Γ\G, ρ, π) f : G Hom C (V ρ,v δ ) 1) γ Γ f(γx)=f(x) ρ(γ) 1, 2) f(x) 2 d Γ\G (ẋ) <, Γ\G 3) k K f(xk) =δ(k 1 ) f(x), 4) ϕ C c (G p /A p,χ p,δ p ) o (p P) G p /A p f(xy 1 )ϕ(y)dμ p (ẏ) = ψ p (ϕ)f(x) (f,g) = (f(x),g(x))d Γ\G (ẋ) Γ\G Hilbert G Hecke C c (G p /A p,χ p,δ p ) o (p P) G p π p 3 Siegel 3.1 [16, Chap.2] 16

118 108 G Lie Lie g g Killging B g (X, Y ) = tr(ad(x) ad(y )) g exp : g G g Cartan θ Lie g 2 g B g (X, θx) (X g) Cartan g = p k p, k θ 1, 1 k G K G G K G- K p G ((k, X) k exp X) θ Aut(g) θ(k exp X) =k exp( X) (k K, X p) G [7, p.480, Cor 1] (ad(h 0 ) p ) 2 = 1 H 0 Z(k) (G, K) Hermite g C = g R C p ± = {X g C ad(h 0 )X = ± 1X} p ± g C Lie p C = p + p k = {X g ad(h 0 )X =0} Lie(G C )=g C Lie G C g C Lie k C, p ± G C K C, P ± P ± exp : p ± P ± p ± [k C, p ± ] p ± P + K C, K C P G C P + K C P G C P + K C P G C (p, k, q) pkq K C P G C i : G G C 1) i d(i) :g g C 2) i(g) P + K C P i 1 (K C P )=K. G/K i i(g)k C P /K C P P + K C P /K C P P + log p + (log = exp 1 ) G/K p + D ġ D= G/K i(g) =expz k q (z p +,k K C,q P ) 17

119 109 ġ = z 1 D= G/K 0 p + dim D =dim R p =dim R p + D p + g G z D p + exp z i(g)k C P i(g)expz i(g)k C P i(g)expz =expg(z) J(g, z) q (g(z) D,J(g, z) K C,q P ) 1) (g, z) g(z) G D K 0 D 2) J : G D K C z D g, h G, z D J(gh, z) =J(g, h(z))j(h, z), 3) k K J(k, 0) = i(k) z D p + k(z) = Ad(i(k))z J K C (δ, V δ ) J δ (g, z) =δ(j(g, z)) 1.2 J δ : G D GL C (V ) G C g g X g C exp X =exp(x) X X g C g. z, z D exp z i(g)k C P exp z i(g)k C P + i(g) P + K C P exp z 1 exp z P + K C P exp z 1 exp z = p K(z,z) 1 q (p P +,K(z,z) K C,q P ) 1) g G K(g(z ),g(z)) = J(g, z )K(z,z)J(g, z) 1, 2) K(z,z)=K(z,z ) 1, 3) z D K(0,z)=1 p C Hermite X, Y = B g (X, Y ) k K C Ad(k)X, Y = X, Ad(k) 1 Y (X, Y p C ) 18

120 110 p + Lebesgue dz dz K C p + Ad p + K Ad (z,z)=ad p +(K(z,z)), J Ad (g, z) =Ad p +(J(g, z)) g G d(g(z)) dg(z) = det J Ad (g, z) 2 dz dz d D (z) =detk Ad (z,z) 1 dz dz (z D) D G- (δ, V δ ) K C K δ(k) =J δ (k, 0) (k K) V δ Hermite (, ) δ k K C (δ(k)u, v) δ =(u, δ(k) 1 v) δ (u, v V δ ) π δ =Ind G Kδ E δ E δ ϕ : G V δ 1) k K ϕ(xk) =δ(k) 1 ϕ(x), 2) ϕ(x) 2 δd G/K (ẋ) < G/K G/K (ϕ, ψ) = (ϕ(x),ψ(x)) δ d G/K (ẋ) G/K ϕ(x) 2 δd G/K (ẋ) =0 ϕ Hilbert G (π δ (g)ϕ)(x) =ϕ(g 1 x) G E δ D K δ (z,z)=δ(k(z,z)) (z,z D) ϕ : D V δ (K δ (z, z) 1 ϕ(z),ϕ(z)) δ d D (z) < D 0 ϕ L 2 (D,δ) (ϕ, ψ) δ = (K δ (z,z) 1 ϕ(z),ψ(z)) δ d D (z) D Hilbert ϕ L 2 (D,δ) ϕ : G V δ ϕ(g) =J δ (g, 0) 1 ϕ(g(0)) (g G) ϕ ϕ Hilbert L 2 (D,δ) E δ π δ =Ind G Kδ L 2 (D,δ) 19

121 111 G L 2 (D,δ) π δ (π δ (g)ϕ)(z) =J δ (g 1,z) 1 ϕ(g 1 (z)) (g G, ϕ L 2 (D,δ)) D ϕ L 2 (D,δ) H δ 1) H δ L 2 (D,δ) G- 2) E H δ G- E = {0} E = H δ H δ {0} (π δ,h δ ) G K- δ (π δ,h δ ) ϕ, ψ H δ (π δ (x)ϕ, ψ) 2 d G (x) < G V δ D H δ,poly H δ K- p + C[p + ] V δ D V δ C C[p + ] K C C[p + ] ρ (ρ(k)f)(x) =f(ad(k) 1 X) n C[p + ] n ρ n K- H δ,poly = δ ρ n n=0 δ ρ 0 = δ n >0 δ ρ n δ δ π δ K 1 π δ K- δ H δ δ- H δ (δ) V δ π δ K- δ Φ πδ,δ(g) =J δ (g 1, 0) 1 (g G) H δ {0} H 0 Z(k) k Cartan t g Cartan t C g C Φ Φ 1t = {α Hom C t C, C) α(k) 1R} 1t (, ) Φ Weyl Φ c = {α Φ α(h 0 )=0} = {α Φ g C,α k C }, Φ ± n = {α Φ α(h 0 )=± 1} = {α Φ g C,α p ± } 20

122 112 (g C,α α Φ ) Φ c k C Φ c Φ + c Φ+ =Φ + c Φ + n Φ K C δ Φ + c λ ρ = 1 α 2 α Φ + H δ {0} (λ + ρ, α) < 0for α Φ + n 3.2 (V,D) V D G = Sp(V )={σ GL R (V ) D(xσ, yσ) =D(x, y) for x, y V } GL R V Lie Sp(V ) Lie g = sp(v )={X gl R (V ) D(xX, y)+d(x, yx) =0for x, y V } Killing B g (X, Y )=(dim R V + 2)tr(XY ) (X, Y g) g I V = {I Sp(V ) I 2 = 1 V,D(xI, x) > 0 for 0 x V } (σ, I) σiσ 1 Sp(V ) I V I I V V 1x = xi (x V ) V I x, y I = D(xI, y)+ 1D(x, y) (x, y V ) V I Hermite U(V I,, I ) Sp(V ) I I V I 0 I V θx = I 0 XI0 1 (X g) g Cartan Cartan g = k p k G K = U(V I0,, I0 ) H 0 = 1 2 I 0 Z(k) (ad(h 0 ) p ) 2 = 1 (G, K) Hermite V C = V R C D (V C,D) G C = Sp(V C ) i : G G C W ± = {x V C xi 0 = ± 1x} 21

123 113 V C = W W + D(W,W )=D(W +,W + )=0, : W W + (x, y) D(x, y) C pairing b Hom C (W,W + ) t b Hom C (W,W + ) x, y t b = xb, y for x, y W d End C (W + ) t d End C (W ) x t d, y = x, yd for x W,y W + c Hom C (W +,W ) t c Hom C (W +,W ) a End C (W ) t a End C (W ) V C = W W + σ End C (V C ) ( ) a b a End C (W ), b Hom C (W,W + ) σ =, c d c Hom C (W +,W ), d End C (W + ) (x, y)σ = (xa + yc, ab + yd) (x W,y W + ) V C V x x W V C f Hom C (W, V C ) ( f ) Hom C (W,V C ) f(x) =f(x) ( (x ) W ) a b d c f = End C (V C ) f = End C (V C ) c d b a {( ) } d c Sp(V )={σ Sp(V C ) σ = σ} = Sp(V C ) c d {( ) } p + 0 b = 0 0 b Sym C(W,W + ), {( ) } p 0 0 = c 0 c Sym C(W +,W ), {( ) } t d 0 k C = 0 d b End C(W + ), Sym C (W,W + )={b Hom C (W,W + ) t b = b}, Sym C (W +,W )={c Hom C (W +,W ) t c = c} 22

124 114 {( ) } P + 1 b = 0 1 b Sym C(W,W + ), {( ) } P 1 0 = c 1 c Sym C(W +,W ) k C G C {( ) t } d 1 0 K C = 0 d d GL C(W + ) x 1 2 (x xi 0 1) V I0 W + V I0 Hermite, I0 W + x, y I0 =2 1D(x, y) = 2 1 y, x (x, y W + ) {( ) } d 0 K = K C Sp(V )= 0 d d U(W +,, I0 ) P + K C P = {( a c ) } b Sp(V C ) d d GL C(W + ) G C = Sp(V C ) G = Sp(V ) G K C P = K 3.1 ( ) I V = G/K p + 0 b D V p + b Sym C (W,W + ) 0 0 ( ) D V Sym C (W,W + a b ) σ = c d Sp(V ) z D V Sym C (W,W + ) ( ) t σ(z) =(az + b)(cz + d) 1 (cz + d) 1 0 D V, J(σ, z) = K C 0 cz + d D V = {z Sym C (W,W + ) 1 W + z z Her + (W + )} Her(W + )={T End C (W + ) xt, y I0 = x, yt I0 for x, y W + }, Her + (W ) = {T Her(W + ) xt, x I0 > 0 for 0 x W + } 23

125 115 ( ) K(z 1 z z 0,z)= 0 (1 zz ) 1 (z,z D V Sym C (W,W + ) (V,D) 2 V = W W σ End C (V C ) V C = W C W C [ ] a b σ = (a End C (W C),b Hom C (W C,W C )etc.) c d pairing, : W W (x, y) D(x, y) R b Hom R (W,W) t b Hom R (W,W) x t b, y = x, yb ( x, y W ) d End R (W ) t d End R (W ) x t d, y = x, yd ( x W,y W ) W ρ = W C W + ρ = W C ρ Sp(V C ) p ± =Ad(ρ 1 )p ± {[ ] } p + 0 b = 0 0 b Sym C(W C,W C ), {[ ] } p 0 0 = c 0 c Sym C(W C,W C) P + = ρ 1 P + ρ, P = ρ 1 P ρ, KC = ρ 1 K C ρ X exp X p + =Ad(ρ)p + P + {[ ] t } d K 1 0 C = 0 d d GL C(W C ), {[ ] } P + 1 b = 0 1 b Sym R(W C,W C ), {[ ] } P + KC P a b = Sp(V C ) c d d GL C(W C ) 2 x, y W D(x, y) =0 W V V Lagrange W V Lagrange dim R W = 1 2 dim R V V Lagrange Sp(V ) Lagrange W, W V V = W W (V,D) Lagrange (W, W ) σ Sp(V ) (W, W ) σ =(Wσ,W σ) 24

126 116 ρsp(v ) P + K C P Sp(V )ρ P + KC P z D V p + exp z Sp(V )K C P exp z ρ Sp(V ) K C P P + KC P exp z ρ =expẑ k q (ẑ p +,k K C,q P ) H V = {ẑ p + z D V } Sym C (W C,W C )} [ ] b Sym C (W C,W 0 b C ) p + z ẑ 0 0 ρ Cayley σ Sp(V ) z H V exp z Sp(V ) K C P σ exp z =expσ(z) J(σ, z) q (σ(z) H V,J(σ, z) K C,q P ) (σ, [ z) ] σ(z) Sp(V ) H V a b σ = Sp(V ) c d σ(z) =(az + b)(cz + d) 1, J(σ, z) = [ t (cz + d) cz + d ] H V = {z Sym C (W C,W C Im z Sym + R (W,W)} Sp(V ) H V Sym + R (W,W)={T Sym R (W,W) x, xt > 0 for 0 x W } z H V W C Hermite v, w z = D(v(Im z) 1, w) (v, w W C ) z H V {σ Sp(V ) σ(z) =z} U(W C,, z ) (σ J(σ, z) WC ). K = {σ Sp(V ) σ( 0) = 0} 3.3 Siegel [6] Q (V Q,D) V Q = W Q W Q V = V Q Q R,W = W Q Q R,W = W Q Q R Γ Sp(V Q ) V Q Z- Λ V Q Γ Sp(Λ) = {σ Sp(V Q ) Λσ =Λ} 25

127 117 3 dim R V > 2 0 <N Z Sp(Λ,N)={γ Sp(Λ) xγ x (mod N Λ) for x Λ} Γ dim R (V )=2 (ρ, V ρ ) Γ [ ] Ker ρ Γ t d 1 0 K C d GL C (W C ) (δ, V δ ) 0 d K C = GL C (W C ) Young m 1 m 2. m n m n > 0 J δ (σ, z) =δ(j(σ, z)) (σ Sp(V ),z H V ) k J δ (k, 0) K V δ Hermite 4 V ρ,v δ Hermite (, ) ρ, (, ) δ T Hom C (V ρ,v δ ) (Tu,v) δ =(u, T v) ρ for u, V ρ,v V δ T Hom C (V δ,v ρ ) Hom C (V ρ,v δ ) Hermite (S, T ) ρ,δ =tr(s T ) F : H V Hom C (V ρ,v δ ) 1) γ Γ F (γz) =J δ (γ,z) F (z) ρ(γ) 1, 2) σ Sp(V Q ) y 0 Sym R (W,W) F σ (z) {z H V Im z y 0 } F Γ ρ δ (δ, ρ)- Siegel J δ (σ, z) =δ(j(σ, z)) 3 G A, B (A : A B) < (B : A B) < A [ B ] 4 a b k = K Im 0 =Imk( 0) = c d t (c 0+d) 1 Im 0 (c 0+d) 1 δ(c 0+d) = δ((c 0+d) 1 )=δ((im 0) 1 t (c 0+d) Im 0) V δ Hermite δ(d) = δ((im 0) 1 t d Im 0) d GL C (W C ) 26

128 118 (σ Sp(V ),z H V ) F σ (z) =J δ (σ, z) 1 F (σz) (z H V ) 2) dim R V>2 1) Koecher (δ, ρ)- Siegel F Sp(V ) f F (σ) =J δ (σ, 0) 1 F (σ( 0)) (σ Sp(V )) Sp(V ) F Γ (δ, ρ)- Siegel z = σ( 0) H V (σ Sp(V )) f F (σ) 2 =(δ((im 0) 1 Im z) F (z),f(z)) ρ,δ F p>0 f(σ) p d Sp(V ) (σ) < Γ\Sp(V ) Satake [14] p>0 pm n 2n Γ (δ, ρ)- Siegel F f F (σ) p d Sp(V ) (σ) < Γ\Sp(V ) F Γ (δ, ρ)- S δ (Γ,ρ) 2.4 Siegel K C K C (k ρkρ 1 ) KC = GL C (W C ) ( ) (δ, V δ ) K C t d 1 0 K C d GL C (W + ) δ GL C (W + ) 0 d Young m 1 m 2. m n K- δ H δ {0} m n >n Satake

129 m n >n K- δ (π δ,h δ ) F f F S δ (Γ,ρ) A δ (Γ\Sp(V ),ρ,π δ ) II Siegel 4 Weil 4.1 R : p =, Z : p =, Q p =, Z p = p- Q p : p< p- Z p : p< Q p Haar dt vol(q p /Z p )=1 :p =, vol(z p )=1 :p< Q p χ (V,D) Q p V Haar d V (x) (x, y) χ(d(x, y)) V Fourier ϕ(y) = ϕ(x)χ( D(x, y))d V (x), ϕ(x) = ϕ(y)χ(d(x, y))d V (y) V H(V )=V Q p (x, s) (y, t) =(x + y, s + t +2 1 D(x, y)) (V,D) Heisenberg H(V ) Z(H(V )) = {(0,t) t Q p } (0,t)=t Q p d H(V ) (x, t) =d V (x)dt H(V ) Haar H(V ) (V,D) Lagrange W V H(V ) W Q p χ W (y, t) =χ(t) (t Q p ) H(V ) (π χ W,Hχ W )=IndH(V ) W Q p χ W H χ W 1) g W Q p ϕ(gh) =χ W (g)ϕ(h), 2) ϕ(h) 2 dḣ< (W Q p )\H(V ) V 28

130 120 ϕ : H(V ) C (ϕ, ψ) = W Q p \H(V ) ϕ(h)ψ(h)dḣ Hilbert H(V ) (π χ W (h)ϕ)(g) = ϕ(gh) W Haar d W (x) (W Q p \H(V ) H(V )- dġ ϕ(h)d H(V ) (h) = dġ d W (x)dtϕ((x, t)h) H(V ) (W Q p )\H(V ) W Q p (ϕ C c (H(V ))) (π χ W,Hχ W ) Heisenberg H(V ) π H(V ) t Q p = Z(H(V )) π(t) =χ(t) Hilbert E H(V ) 1 E π 1 E π χ W π H(V ) t Q p = Z(H(V )) π(t) =χ(t) π π χ W V Lagrange W, W V (π χ W,Hχ W ) (πχ W,H χ W ) H(V ) W W Haar d W W Haar W/(W W ) W /(W W ) d W/(E W ) = d W /d W W, d W /(W W ) = d W /d W W 5 W /(W W ) (W /(W W )) Haar d (W /(W W )) d W /(W W ) Q p - g W,W : W/(W W ) (W /(W W )) ẇ,g W,W (ẇ) = D(w, w ) 0 < g W,W R d (W /(W W )(g W,W (ẇ)) = g W,W d W/(W W )(w) 5 G H G, H Haar d G (g),d H (h) G/H G- d G/H (ġ) ϕ(g)d ( g)= d G/H (ġ) d H (h)ϕ(gh) (ϕ C c (G)) G G/H d G/H = d G /d h H 29

131 121 g W,W 1/2 d W /(W W ) W W Haar d W W T χ W,W : Hχ W Hχ W (T χ W,W ϕ)(h) = ϕ((w, 0)h) g W,W 1/2 d W /(W W )(ẇ ) W /(W W ) (ϕ H χ W,c ) Hχ W,c ϕ : H(V ) C 1) g W Q p ϕ(gh) =χ W (g)ϕ(h), 2) ḣ ϕ(h) W Q p\h(v ) H χ W 1) T χ W,W T χ W,W = T χ W,W = id, 2) h H(V ) T χ W,W πχ W (h) =πχ W T χ W,W Q p - X X x X, α X x, α = α(x) V X = X X D X ((x, α), (y, β)) = x, β y, α X, X V X (V X,D X ) Lagrange V = X X σ End Qp (V X ) ( ) a b σ = Q X Q : X Q p c d 1) λ Q p,x X Q(λx) =λ 2 Q(x), 2) (x, y) Q(x, y) =Q(x +y) Q(x) Q(y) (x, y X) Q p - Q SymQp (X, X ) x, y Q = Q(x + y) Q(x) Q(y) (x, y W ) ( ) 1 Q σ = Sp(V X ) W Q = Xσ V X Lagrange T χ X,W Q 0 1 T χ W Q,X T χ X,X (π χ X,H χ X ) (π χ X,Hχ X ) T χ X,W Q T χ W Q,X = γ χ (Q) T χ X,X γ χ (Q) C 1 Q Weil X Q rad(q) ={x X x, y Q =0for y X} 6 T χ W,W 4.2 [12] 30

132 122 X/rad(Q) Q reg (ẋ) =Q(x) γ χ (Q) =γ χ (Q reg ) Q γ χ (Q) Q p Witt W Qp C 1 Q p (V,D) Lagrange W i V (i =1, 2, 3) W 1 W 2 W 3 Q W1,W 2,W 3 Q W1,W 2,W 3 (w 1,w 2,w 3 )=D(w 1,w 2 )+D(w 2,w 3 )+D(w 3,w 1 ) (w i W i ) rad(q W1,W 2,W 3 )={(w 1,w 2,w 3 ) w 1 w 2 W 3,w 2 w 3 W 1,w 3 w 1 W 2 } (1, 2, 3) (i 1,i 2,i 3 ) ( ) Q Wi1,W i2,w i3 =sign Q W1,W 2,W 3 i 1 i 2 i Lagrange W i V (i =1, 2, 3) T χ W 1,W 2 T χ W 2,W 3 T χ W 3,W 1 = γ χ (Q W1,W 2,W 3 ) id. 4.3 Q p (V,D) V = W W x W,y W x, y = D(x, y) W,W Haar d W (x),d W (y) (x, y) χ( x, y ) W Fourier ϕ(y) = ϕ(x)χ( x, y )d W (x), ϕ(x) = ϕ(y)χ( x, y )d W (y) W W d V (x, y) =d W (x)d W (y) ϕ L 2 (W ) ϕ(g) =χ ( ν (t +2 1 x, y ) ) ϕ(x) (g =((x, y),t) H(V )) ϕ ϕ L 2 (W ) H χ W Ind H(V ) W Q p χ W L 2 (W ) Π χ (h =((x, y),t) H(V ) ϕ L 2 (W ) (Π χ (h)ϕ)(w )=χ ( t + w,y +2 1 x, y ) ϕ(w + x) (w W ) H(V ) (Π χ,l 2 (W )) Schrödinger H(V ) W Q p χ W (x, t) =χ(t) 1 H(V ) Ind H(V ) W Q p χ W L 2 (W ) ˇΠ χ h =((x, y),t) H(V ) ϕ L 2 (W ) ( ˇΠ χ (h)ϕ)(w) =χ ( t x, w +2 1 x, y ) ϕ(w y) 31

133 123 ˇΠχ Π χ, χ : L 2 (W ) L 2 (W ) C ϕ L 2 (W ) L 1 (W ), ψ L 2 (W ) L 1 (W ) ϕ, ψ χ = d W (w ) d W (w)ϕ(w )ψ(w)χ( w,w ) W W h H(V ), ϕ L 2 (W ), ψ L 2 (W ) Π χ (h)ϕ, ψ χ = ϕ, ˇΠ χ (h 1 )ψ χ ( ˇΠ χ,l 2 (W )) Schrödinger σ Sp(V ) h =(x, t) H(V ) h σ =(xσ, t) h h σ H(V ) σ Sp(V ) Π σ χ(h) =Π χ (h σ )(h H(V)) (Π σ χ,l 2 (W )) H(V ) t Q p = Z(H(V )) Π σ χ(t) =χ(t) Π σ χ Π χ h H(V ) T 1 Π χ (h) T =Π χ (h σ ) T Aut(L 2 (W )) Aut(L 2 (W )) ϕ L 2 (W ) T Tϕ Aut(L 2 (W )) Hausdorff Sp(V ) Aut(L 2 (W )) Mp(V )={(σ, T ) T 1 Π χ (h) T =Π χ (h σ )for h H(V )} ϖ : Mp(V ) Sp(V ) ((σ, T ) σ) Mp(V ) Sp(V ) C 1 Mp(V [ ) ] a GL Qp (W a 0 ) d(a) = 0 a 1 Sp(V ) d 0 (a) Aut(L 2 (W ) (d 0 (a)ϕ)(w )= det a 1/2 ϕ(w a) (ϕ L 2 (W ),w W ) d(a) =(d(a), d 0 (a)) Mp(V ) d : GL Qp (W ) Mp(V ) [ ] b Sym Qp (W 1 b,w) t(b) = 0 1 Sp(V ) t 0 (b) Aut(L 2 (W )) (t 0 (b)ϕ)(w )=χ(2 1 w,w b ) ϕ(w ) (ϕ L 2 (W ),w W ) 32

134 124 t(b) = (t(b), t 0 (b)) Mp(V ) t :Sym Qp (W,W) Mp(V ) [ R- ] c : W W d (c) = 0 t c 1 Sp(V ) d 0(c) Aut(L 2 (W )) c 0 (d 0ϕ)(w )= c 1/2 ϕ(w t c 1 ) (ϕ L 2 (W )),w W prime ) d (c) =(d (c), d 0(c)) Mp(V ) d W (wc) = c d W (w) 0 < c R c d (c) {[ ] } a b Ω(V )= Sp(V ) c d c : W W : σ = [ a c ] b Ω(V ) σ =t(ac 1 )d (c)t(c 1 d) d r 0 (σ) =t 0 (ac 1 ) d 0(c) t 0 (c 1 d) Aut(L 2 (W )) r(σ) =(σ, r 0 (σ)) Mp(V ) r :Ω(V ) Mp(V ) (σ, λ) (σ, λr(σ)) Ω(V ) C 1 Mp(V ) ϖ 1 (Ω(V )) Mp(V ) [19, p.186] Φ:Mp(V ) C 1 1) [ λ ] C 1 Φ(1,λ)=λ 2, a b 2) σ = Ω(V ) Φ(r(σ)) = ( c, 1) 2 γ χ (Q 1 ) dim V c d Q 1 (x) =x 2 Q p (a, b) 2 Hilbert 7 Sp(V )=KerΦ Mp(V ) ϖ : Sp(V ) Sp(V ) ((σ, T ) σ) 7 a, b Q p {[ ] } s + t a u+ v a H = b(u v a) s t s, t, u, v Q a p M 2 (Q p ( a)) Q p - M 2 (Q p ) Q p { 1 : H M 2 (Q p ), (a, b) 2 = 1 : H Q p Hilbert 33

135 125 {(1, ±1)} ω χ : Sp(V ) Aut(L 2 (W )) ((σ, T ) T ) Sp(V ) Weil a GL Qp (W ) cw W d (c)d(a) =d (ca) Φ(d(a)) = (det a, 1) 2 η χ (a) =γ χ (Q 1 )γ χ ( det a Q 1 ) η χ (a) 2 =(deta, 1) 2 d(a) =(d(a),η χ (a) 1 d(a)) Sp(V ) (3) 4.4 Z p - L W 8 L = {x W χ( x, L ) =1} Λ=L L H(Λ) = Λ Q p H(V ) χ Λ : H(Λ) C 1 (((x, y),t) χ (t + 12 ) x, y ) H(Λ) π χλ =Ind H(V ) H(Λ) χ Λ 1) λ H(Λ) θ(λg) =χ Λ (λ)θ(g), 2) θ(g) 2 d(ġ) < H(Λ)\H(V ) H(V ) θ h H(V ) θ Ind H(V ) H(Λ) χ Λ (π χλ (h)θ)(g) =θ(gh) Schwartz 9 ϕ S(W ) Θ ϕ (h) = ϕ(x+l)χ (t + 12 ) x, y + l, y d L (l) (h =((x, y),t) H(V )) L H(V ) Θ ϕ Θ ϕ Ind H(V ) H(Λ) χ Λ Θ ϕ = ϕ ϕ Θ ϕ H(V ) (Π χ,l 2 (W )) Ind H(V ) H(Λ) χ Λ 8 W Z p - W Q p 9 p = p < 34

136 126 Ind H(V ) H(Λ) χ Λ Schrödinger Λσ =Λ σ Sp(V ) Sp(Λ) Sp(Λ) Sp 0,χ (Λ) = {γ Sp(Λ) χ Λ (λ γ )=χ Λ (λ) for λ H(Λ)} γ Sp 0,χ (Λ) Hilbert Ind H(V ) H(Λ) χ Λ r χ (γ) (r χ (γ)θ)(h) =θ(h γ ) (θ Ind H(V ) H(Λ) χ Λ,h H(V )) ϕ Θ ϕ r χ (γ) L 2 (W ) r χ γ (γ,r χ (γ)) Sp 0,χ (Λ) Mp(V ) Sp0,χ (Λ) = ϖχ 1 Sp 0,χ (Λ) ρ χ = ρ Λ,χ : Sp 0,χ (Λ) C 1 γ Sp 0,χ (Λ) ω χ ( γ) =ρ χ ( γ) r χ (γ) (γ = ϖ χ ( γ)) a V H(Λ) χ Λ,a (λ) =χ Λ (λ) χ(d(a, l)) (λ =(l, t) H(Λ)) a =(a,a ) V = W W W Schwartz ϕ S(W ) ϑ ϕ [a]( g) =Θ ωχ,j ( g)ϕ(a, 0) ( ) 1 = χ 2 a,a (ω χ,j ( g)ϕ)(a + l)χ( l, a )d L (l) L ( g Sp(V ) J ) γ Sp 0,χ (Λ) λ H(Λ) ϑ ϕ [a](( γ,λ) g) =ρ Λ ( γ)χ Λ,aγ (λ)ϑ ϕ [aγ]( g). [ ] ω χ ( γ) Π χ (λ) Ind H(V ) H(Λ) χ Λ ψ = ω χ,j ( g)ϕ ϑ ϕ [a](( γ,λ) g) =(ω χ ( γ) Π χ (λ)θ ψ )(a, 0) = ρ Λ ( γ)(π χ (λ)θ ψ )(aγ, 0) = ρ Λ ( γ)θ ψ ((aγ, 0)λ). 35

137 127 λ =(l, t) (x, s) H(V ) (l, t) 1 (x, s)(l, t) =( l, t)(x + l, s + t +2 1 D(x, l)) =(x, s +2 1 D(x, l) 2 1 D(l, x + l)) =(0,D(x, l))(x, s) Θ ψ ((aγ, 0)λ) =Θ ψ (λ(0,d(aγ, l))(aγ, 0) = χ Λ (λ(0,d(aγ, l)))θ ψ (aγ, 0) g = σ Sp(V ),a= W Schwartz ϕ S(W ) ϑ ϕ ( σ) = (ω χ ( σ)ϕ)(l)d L (l) ( σ Sp(V )) L γ Sp 0,χ (Λ) ϑ ϕ ( γ σ) =ρ Λ ( γ) ϑ ϕ ( σ). 4.5 GL Qp (V ) GSp(V )={σ GL Qp (V ) D(xσ, yσ) =ν(σ) D(x, y) for x, y V } σ GSp(V ) h =(x, t) H(V ) h σ =(xσ, ν(σ)t) H(V ) h h σ H(V ) Sp(V ) J = Sp(V ) H(V ) Jacobi Sp(V ) ϖ : Sp(V ) Sp(V ) H(V ) Sp(V )J = Sp(V ) H(V ) ϖ J : Sp(V ) J Sp(V ) J (( σ, h) (ϖ( σ),h)) Sp(V ) Weil (ωχ,l 2 (W )) Sp(V )J (ω χ,j,l 2 (W )) ω χ,j ( σ, h) =ω χ ( σ) Π χ (h) Sp(V ) π Sp(V )J Sp(V ) Sp(V )J π J ρ = π J ω χ,j Sp(V )J t Q p = Z(H(V )) = 36

138 128 Z( Sp(V ) J ) ρ(t) =χ(t) t Q p ρ(t) =χ(t) Sp(V )J (ρ, H) H(V ) Hilbert E H = E L 2 (W ) ρ H(V ) = 1 E Π χ E = L H(V ) (L 2 (W ),H) T = E Sp(V ) π sup Tϕ / ϕ Hilbert 0 ϕ L 2 (W ) π( σ)t = ρ( σ, 1) T ω χ ( σ) 1 ρ = π J ω χ,j π π J ω χ,j Sp(V ) π ρ Z( Sp(V )J = χ Sp(V )J ρ π J ω χ,j π 5 1/2 5.1 (V,D) 7.1 e(t) =e (t) = exp(2π 1t) g =(σ, h) Sp(V ) J (σ Sp(V ),h =(x, t) H(V )) Z =(z, w) H V,J [ ] η(g; Z) =e (t + 12 D(x, x z J(σ, z) 1 σ) 1 W +D(x, wj(σ, z) 1 σ)+ 1 ) 2 D(w, wj(σ, z) 1 σ) Z =(z,w ) H V,J ( ) 1 κ(z ; Z) =e 2 (w w)(z z) 1,w w κ(g(z ),g(z)) = η(g; Z )κ(z ; Z)η(g; Z) z H V w,w W C κ z (w,w)=κ(z,w ; z,w) Heisenberg H(V ) Z(H(V )) = R χ(t) =e 2π 1t H(V ) π χ =Ind H(V ) Z(H(V )) χ 1) t R ϕ(g(0,t)) = χ(t) 1 ϕ(g) =e(t)ϕ(g), 2) ϕ(g) 2 d(ġ) < H(V )/Z(H(V )) 37

139 129 H(V ) ϕ h H(V ) (π χ (h)ϕ)(g) =ϕ(h 1 g) H V,J (z,0) H V,J H(V )- Ω z = {z} W C (z,w) =w Ω z = W C (z,0) Ω z H(V ) Z( H(V )) Ind H(V ) Z(H(V )) χ Ω z = W C d z (w) =(detimz) 1 d WC (w) Ω z = W C H(V )- g Sp ( V ) J g(z,w) = (z,w ) H V,J d z (w )=d z (w) ϕ Ind H(V ) Z(H(V )) χ Ω z = W C ϕ ϕ(w) =η(h; z,0) 1 ϕ(h) (w = h(0) Ω z = W C,h H(V )) ϕ(h) 2 = ϕ(w) 2 κ z (w, w) h H(V ) ψ = π χ (h)ϕ z H V ψ(w) =η(h 1 ; z,w) ϕ(h 1 (w)) (w Ω z = W C ) Ω z = W C ϕ W C ϕ(w) 2 κ z (w, w)d z (w) < Hilbert L z H(V ) L z π z (π z (h)ϕ)(w) =η(h 1 ; z,w)ϕ(h 1 (w)) (h H(V ),ϕ L z ) Ω z = W C ϕ L z H z L z (π z, L z ) (π z, H z ) (π z, H z ) H(V ) Schrödinger ( ˇΠ e,l 2 (W )) Re T =(T + T )/2 T Sym C (W C,W C ) S C det 1/2 det 1/2 T = e π w,wt d W (w) W (T S C ) (det 1/2 T ) 2 =dett 1 d W (w) = n dx i (w = i=1 n x i u i ) W R- {u i } 1=1,,n det T =det( u i,u j T ) i,j=1,,n T Sym R (W,W) det 1/2 T = i=1 38

140 130 (det T ) 1/2 n det n/2 T =(det 1/2 T ) n Re S =(S + S)/2 S Sym C (W C,W C) S C det 1/2 det 1/2 S = W e π ws,w d W (w) (S S C) T T 1 S C S C det 1/2 (T 1 )= det 1/2 T z H V γ(z) =det 1/2 (z/ 1) det(2im z) 1/4, ( q z (w) =e 1 ) 2 wz 1,w (w W C ) ψ L 2 (W ) Q z (ψ)(w) =γ(z) q z (w v)ψ(v)d W (v) (w W C ) W ψ Q z (ψ) H(V ) ( ˇΠ e,l 2 (W )) (π z, H z ) (π z, H z ) Schrödinger ( ˇΠ e,l 2 (W )) Fock (Π e,l 2 (W ) Fock Fourier F : L 2 (W ) L 2 (W ) (Fϕ)(w) = ϕ(v)e( v, w )d W (v) (ϕ L 2 (W ) L 1 (W )) W [ ] z H V z 1 0 = z H V ε = 0 1 GSp(V ) H(V ) H z ˇπ z (h) =π z (h ε )(h H(V )) L 2 (W ) F L 2 (W ) Q z H z (Π e,l 2 (W )) (ˇπ z, H z ) κ z (w, w) = κ z (w, w) Hiulbert H z H z 5.2 Heisenberg H(V ) (π z, H z ) ϕ H z h H(V ) w Ω z = W C w = h(w) Ω z = W C (π z (h)ϕ)(w )=η(h; z,w) 1 ϕ(w) g =(σ, h) Sp(V ) J w Ω z = W C g(z,w) = (σ(z),w ) Ω σ(z) = W C (T z (g)ϕ)(w )=η(g; z,w) 1 ϕ(w) (ϕ H z ) 39

141 131 T z (g) H z H σ(z) g Sp(V ) J T z (g g)=t σ(z) (g ) T z (g) T z (g) Aut(L 2 (W )) L 2 (W ) T z (g) Q z L 2 (W ) Q σ(z) Hz T z (g) H σ(z) z,z H V H z H z U z,z = Q z Q 1 z ϕ H z (U z,zϕ)(w )=γ(z,z) κ(z,w ; z,w) 1 ϕ(w)κ z (w, w)d z (w) W C γ(z,z)=γ(z )γ(z)det 1/2 { (z / 1) 1 + (z/ 1) 1} ( z =det 1/2 ) z 2 det(im z ) 1/4 det(im z) 1/4 1 T z (g) Aut(L 2 (W )) Q z : L 2 (W ), H z H z T z (g) =U z,σ(z) T z (g) Aut(H z ) (g =(σ, h) Sp(V ) J ) Ť z (g) Aut(L 2 (W )) L 2 (W ) Ť z (g) F L 2 (W ) Q z H z T z (ε 1 gε) L 2 (W ) F L 2 (W ) Q σ(z) H σ(z) g =(σ, h),g =(τ,h ) Sp(V ) J Ť z (g) Ťz(g )=β z (σ, τ)ťz(gg ) σ Sp(V ) z,z H V ( ) ε(σ; z,z)=det 1/2 σ(z ( ) σ(z) z 2 det 1/2 ) z det J(σ, z ) 1/2 det J(σ, z) 1/2 40

142 132 σ, τ Sp(V ) z H V β z (σ, τ)=ε(σ; z,τ(z)) C 1 g, g,g Sp(V ) J (Ťz(g) Ťz(g )) Ťz(g )=Ťz(g) (Ťz(g ) Ťz(g )) σ, τ, δ Sp(V ) β z (τ,δ)β z (στ, δ) 1 β z (σ, τδ)β z (σ, τ) 1 =1 β z C 1 Sp(V ) 2-cocycle Sp(V ) C 1 Mp(V ; z) Mp(V ; z) =C 1 Sp(V ) (ε, σ) (η, τ) =(εηβ z (σ, τ),στ) Mp(V ; z) Lie σ Sp(V ) α z (σ) =detj(σ, z)/ det J(σ, z) 10 σ, τ Sp(V ) β z (σ, τ) 2 = α z (τ)α z (στ) 1 α z (σ) (ε, σ) ε 2 α z (σ) Mp(V ; z) C 1 Sp(V ; z) Sp(V ; z) ={(ε, σ) C 1 Sp(V ) ε 2 = α z (σ) 1 } Mp(V ; z) Lie ϖ z : Sp(V ; z) Sp(V ) ((ε, σ) σ) Sp(V ; z) Sp(V ) (ε, σ) (σ, εťz(σ)) Mp(V ; z) Mp(V ) (ε, σ) Mp(V ; z) Φ(σ, εťz(σ)) = ε 2 α z (σ) t 10[ d 1 ] [ ] 0 0 d K a b C d GL C (W C ) 26 σ = Sp(V ) c d [ t (cz + d) J(σ, z) = 1 ] 0 = cz + d 0 cz + d K C = GL C (W C ) 41

143 133 (ε.σ) (σ, εťz(σ)) Sp(V ; z) Sp(V ) Weil ω χ : Sp(V ; z) Aut(L 2 (W )) ((ε, σ) εťz(σ)) χ(t) =e(t) (ˇω χ,l 2 (W )) ˇω χ ( σ) = ε 1 T z (σ) ( σ =(ε, σ) Sp(V ; z)) 5.3 z 0 = 0 H V Sp(V )= Sp(V ; z0 ) ϖ : Sp(V ) Sp(V ) ((ε, σ) σ), ω : Sp(V ) Aut(L 2 (W )) ((ε, σ) ε Ťz 0 (σ)) Sp(V ) ϕ HV K = ϖ 1 (K) ={(ε, k) C 1 K ε 2 =detj(k, z 0 ) 1 } z 0 H V C 1 K σ =(ε, σ) Sp(V ) J 1/2 ( σ, z) =ε 1 ε(σ; z,z 0 ) det J(σ, z) 1/2 J 1/2 ( σ, z) 2 =detj(σ, z) J 1/2 ( σ, z) σ Sp(V ) z H V σ, τ Sp(V ) J 1/2 ( σ τ,z)=j 1/2 ( σ, τ(z))j 1/2 ( τ,z) det 1/2 : K C 1 ( k =(ε, k) J( k, z 0 )=ε 1 ) K 1 Weil (ω, L 2 (W )) K L 2 (W ) F L 2 (W ) Q z 0 H z 0 Weil H z 0 W C P C[W C ] ϕ(w) =P (w)κ z 0 (w, 0) 1 (w W C ) ϕ H z 0 σ =(ε, σ) Sp(V ) (ω( σ)ϕ)(w) =ε P (wj(σ, z 0 ))κ σ(z0 ) (w, 0) 1 (w W C ) Weil ω K ω K = det 1/2 Sym n n=0 42

144 134 Sym n K = U(W C,, z0 ) n GL C (W C ) Young n K- det 1/2 L 2 (W ) ϕ = F 1 Q 1 z κ 0 z 0 (, 0) 1 L 2 (W ) ( ) 1 ϕ(u) = det(2im z 0 ) 1/4 e 2 u, uz 0 (u W ) g =( σ, h) Sp(V ) J ( σ =(ε, σ) Sp(V ),h H(V )) g =(σ, h) Sp(V ) J g(z 0, 0) = (z,w) H V,J (ω χ,j ( g)ϕ)(u) =ε(ťz 0 (g)ϕ)(u) = ε η(g; z 0, 0) det(2im z) 1/4 e ( ) 1 u, uz + u, w 2 (u W ) ω χ,j ( g) Ind H(V ) H(Λ) χ Λ Θ ϕ Ind H(V ) H(Λ) χ Λ ( (ω χ,j ( g)θ ϕ )(h )=ε η(g; z 0, 0) det(2im z) 1/4 e t 1 ) 2 a,a ϑ[a](z,w) h =(a, t) H(V )(a =(a,a ) W W ) ϑ[a](z,w) = ( ) 1 e 2 l + a, (l + a )z + l + a,w+ a l L (4) Riemenn χ Λ (h) =e (t + 12 ) x, y (h =((x, y),t) H(Λ)) Sp 0 (Λ) = {γ Sp(Λ) χ Λ (h γ )=χ Λ (h) for h H(Λ)} γ Sp 0 (Λ) ϖ( γ) =γ γ Sp 0 (Λ) ϑ [a](γ(z),wj(γ,z) 1 )=ρ Λ ( γ)j 1/2 ( γ,z)η(γ; z,w) 1 ϑ [aγ](z,w) ( ϑ [a](z, w) =e 1 ) 2 a,a ϑ[a](z,w) (x, y) Λ h =((x, y), 0) H(Λ) ϑ[0](z, w + xz + y) =χ Λ (h)η(h; z,w) 1 ϑ[0](z,w) 43

145 ϑ(z) =ϑ[0](z, 0) = ( ) 1 e l, lz 2 l L (z H V ) γ Sp 0 (Λ) ϑ(γ(z))/ϑ(z) =ρ Λ ( γ)j 1/2 ( γ,z) γ Sp 0 (Λ) ρ Λ ( γ) 8 =1 [11] Chap.V, Γ Sp(V Q ) dim R (V )=2 0 <N Z Sp(Λ,N) Γ Sp(V ) Γ =ϖ 1 (Γ) [ ] α Ker α Γ t d 1 0 K C d GL C (W C ) 0 d (δ, V δ ) KC = GL C (W C ) Young m 1 m 2. m n m n > 0 K δ det 1/2 k =(ε, k) J1/2 ( k, z 0 ) 1 J δ (k, z 0 )=ε J δ (k, z 0 ) 3.3 Siegel F : H V V δ 1) γ Γ F (γ(z)) = α( γ) 1 J 1/2 ( γ,z) 1 J δ (γ,z)f(z), 2) σ Sp(V Q ) y 0 Sym R (W,W) det J(σ, z) 1/2 J δ (σ, z) 1 F (σ(z)) {z H V Im z y 0 } F Γ δ det 1/2 α Siegel Sp(V ) f F ( σ) =J 1/2 ( σ, z 0 )J δ (σ, z 0 ) 1 F (σ(z 0 )) ( σ Sp(V ),ϖ( σ) =σ) 44

146 136 Sp(V ) F f F ( σ) = det J(σ, z 0 ) 1/2 J δ (σ, z 0 )F (σ(z 0 )) ϖ( σ) =σ Sp(V ) p>0 f F ( σ) p d Sp(V ) (σ) < Γ\Sp(V ) Satake p>0 p(m n 1/2) 2n Γ δ det 1/2 α Siegel F f F ( σ) p d Sp(V ) (σ) < Γ\Sp(V ) F Γ δ det 1/2 α S δ det 1/2(Γ,α) 3.3 Siegel p =2 Sp(V ) δ det 1/2 Siegel H V (δ det 1/2 (Im z)ϕ(z),ϕ(z)) δ d frakhv (z) < ϕ : H V V δ H δ det 1/2 (ϕ, ψ) = (δ det 1/2 (Im z)ϕ(z),ψ(z)) δ d frakhv (z) H V Hilbert Sp(V ) Hδ det 1/2 π δ det 1/2 (π δ det 1/2( σ)ϕ)(z) =J 1/2 ( σ 1,z) 1 J δ (σ 1,z) 1 ϕ(σ 1 (z)) H δ det 1/2 {0} m n >n (π δ det 1/2,H δ det 1/2) Sp(V ) K- δ det 1/ m n >n F f F Hilbert S δ det 1/2(Γ,α) A δ det 1/2( Γ\ Sp(V ),α,π δ det 1/2) F S δ det 1/2(Γ,α) β V δ θ F β ( σ) =(dimδ) 1/2 β,f F ( σ) ( σ Sp(V )) F β θ F β Hilbert S δ det 1/2(Γ,α) C V δ L 2 ( Γ\ Sp(V ),α 1 ;ˇπ δ det 1/2, ˇδ det 1/2 ) 45

147 V Z- Λ V Q : V V C x, y V Q( 1x, y) = 1Q(x, y), (Q(x, y) Q(y, x) 1R Q V Hermite H Q (x, y) = (Q( 1x, y) Q(x, 1y)) (x, y V ) V Hermite D Q (x, y) =ImH Q (x, y) = 1 1 (Q(x, y) Q(y, x)) (x, y V ) V α :Λ C 1 = {z C z =1} u, v Λ α(u + v) =α(u)α(v)e π 1D Q (u,v) α Hermite Q D Q Hermite Q u, v Λ D Q (u, v) Z V Hermite Q α J Q,α (u, x) =α(u)exp (πq(x, u)+ π ) 2 Q(u, u) (u Λ,x V ) u, v Λ J Q,α (u + v, x) =J Q,α (u, x + v)j Q,α (v, x) V θ u Λ θ(x + u) =J Q,α (u, x)θ(x) θ Λ (Q, α) L(Q, α) V Hermite Λ Q = {x V D Q (u, x) Z for u Λ} Λ Λ Q V 1) (Λ Q :Λ)<, 2) V D Q, 46

148 138 3) V Hermite H Q V Hermite Q α (Λ Q :Λ) 1/2 : H Q, dim C L(Q, α) = 0 : H Q. (V,D) V = W W x W,y W x, y = D(x, y) L,L Z Z- L W,L W L D = {x W x, L Z}, L D = {y W L,y Z} L L D W L L D W Z- (4) Riemann ϑ[a](z,w) = ( ) 1 e l + λ, (l + λ)z + l + λ, w + μ 2 l L (a =(λ, μ) W W, z H V, w W C ) a =(λ, μ) W W z H V ϑ[a](z, ) W C Z- Λ z = {uz + v u L,v L} (Q z,α) Q z = 2 1 x(im z) 1, Im y (x, y W C ) W C Hermite α α(u) =e 2π 1 u,μ, α(v) =e 2π 1 λ,v (u L,v L) Q z h =((x, y), 0) H(V )(x W,y W ) η(h; z, w) =J Qz,1(xz + y, w) L, 1 L Q z θ 1,θ 2 L(Q z,α) θ 1 (w)θ 2 (w)κ z (w, w) Λ z - L(Q z,α) Hermite (θ 1,θ 2 )= θ 1 (w)θ 2 (w)κ z (w, w)d z (w) W C /Λ z

149 a =(λ, μ) W W z H V {ϑ[a+(v, 0)](z, )} v L D /L L(Q z,α) v, v L D (ϑ[a +(v, 0)](z, ),ϑ[a +(v, 0)](z, )) (L D : L ) det(2im z) 1/2 : v v (mod L ), = 0 : v v (mod L ) (V,D) V = W W z 0 H V Sp(V ) Sp(V )= Sp(V ; z0 ) Weil (ω, L 2 (W )) χ(t) =e(t) (G, H) V reductive dual pair K G, L H ϖ : Sp(V ) Sp(V ) K = ϖ 1 (K) G = ϖ 1 (G), L = ϖ 1 (L) H = ϖ 1 (H) G H Sp(V ) ω( G),ω( H) L 2 (W ) von Neumann A G = ω( G), A H = ω( H) 11 Howe [9] A H = A G A G = A H i : G H Sp(V )((g, h) gh) ω G H = ω i G H L 2 (W ) ) (ω G H) disc 1 2) G π π π ω G H H 1 (ω G H) disc = π λ π λ λ Λ (π λ,π λ G, H πλ,π λ H πλ,h π λ H πλ H π λ L 2 (W ) π λ π λ- L 2 (W ) λ 11 Hilbert X L(X) S S = {T L(X) T S = S T for S S} S =(S ) L(X) C- A A = A A X von Neumann 48

150 140 U λ π π ω G H G, H π, π K, L δ, δ π K,π L 1 Z- L W Λ=L L L = {x W x, L Z} Γ Sp 0 (Λ) G, Γ Sp 0 (Λ) H Γ, Γ W Schwartz ϕ S(W ) ϑ ϕ ( σ) = l L (ω( σ)ϕ)(l) ( σ Sp(V )) ϑ ϕ ( γ σ) =ρ Λ ( γ)ϑ ϕ ( σ) ( γ Sp 0 (Λ)) ρ G = ρ Λ Γ,ρ H = ρ Λ Γ f L 2 ( Γ\ G, ρ 1 G ;ˇπ, ˇδ) ϕ S(W ) F f,ϕ (h) = f(g)ϑ ϕ (gh)d G(g) (h H) Γ\ G F f,ϕ ( γh)=ρ H ( γ) F f,ϕ (h) for γ Γ ϕ = U λ (u v) (u H πλ,v H π λ ) F f,ϕ 0 π λ = π π λ = π u H π (δ) ψ C c ( H,δ ) o F f,ϕ (hy)ψ(y)d H(y) = ψ π,δ (ψ) H v H π (δ ) [ ] ψ C c ( G, δ) o ( ) F f,ω(ψ)ϕ (h) = f(g) ψ(x)ϑ ϕ (ghx)d G(x) d G(g) Γ\ G G = d G(g) d G(x)f(gx 1 )ψ(x)ϑ ϕ (gh) Γ\ G G = ψˇπ,ˇδ(ψ ) F f,ϕ (h) = ψ π,δ (ψ) F f,ϕ (h). 49

151 141 u H πλ (δ) ω(ψ)ϕ = U λ (π λ (ψ)u v) = U λ (π λ (ψ e δ )u v) =0 ψ C c ( G, δ) o ψ π,δ (ψ) F f,ϕ = F f,ω(ψ)ϕ =0, F f,ϕ =0 u H πλ (δ) = r Vδ, u = r u i =. i=1 u 1 u r (u i V δ ) ψ C c ( G, δ) o A ψ M r (C) u 1 π λ (ψ). = A ψ u 1. u r u r ψ A ψ C c ( G, δ) o M r (C) C- r ϕ i = U λ (u i v) S(W ) ϕ = ϕ i F f,ϕ1. F f,ϕr ψ C c ( G, δ) o ψ π,δ (ψ) F f,ϕ1. = 0 F f,ω(ψ)ϕ1. = A ψ i=1 F f,ϕ1. F f,ϕr F f,ω(ψ)ϕr F f,ϕr r =1 ψ C c (Ĝ, δ)o ψ πλ,δ(ψ) =A ψ = ψ π,δ (ψ) π λ = π ψ C c (Ĥ,δ ) o F f,ϕ (hy)ψ(y)dĥ(y) = d G(g) d H(y)f(g)ϑ ϕ (ghy) Ĥ Γ\ G H = f(g)ϑ ω(ψ)ϕ (gh)d G(g) Γ\ G ψ π =,δ (ψ) F f,ϕ(g) : v H π (δ ), 0 : v H π (δ ) 50

152 142 H F f,ϕ (hy)ψ(y)d H(y) = ψ π,δ (ψ) F f,ϕ(h) for ψ C c ( H,δ ) o v H π (δ ) F f,ϕ Γ \ H F f,ϕ L 2 ( Γ \ H,ρ H ; π,δ ) f (h) 2 d H(ḣ) Γ \ H = Γ Γ\ G G d G G(ġ, ġ )f(g)f(g ) ϑ ψ (gh)ϑ ψ (g h)d H(ḣ) Γ \ H ϑ ψ (gh)ϑ ψ (g h)= (l,l ) L L (ω χ (gh)ψ)(l)(ω χ (g h)ψ)(l ) Sp(V V ) reductive dual pair (H H, G G) theta H H Δ(H) seesaw dual pair (Δ(H), G) Siegel-Weil ϑ ψ (gh)ϑ ψ (g h)d H(ḣ) =E G((g, g ),λ):g Eisenstein Γ \ H H H G Δ(H) G G [5] Rankin-Selberg f (h) 2 d H(ḣ) = f(g)f(g )E G ((g, g ),λ)d G G(ġ, ġ ) Γ \ H Γ Γ\ G G f L- 6.2 H π H π L 2 (W ) π π - U u H π (δ) =V δ, v H π (δ )=V δ U(u v) S(W ) v V δ u, Θ v (s) = ϑ U(u v) (s) u V δ,s Sp(V ) Θ v : Sp(V ) V δ 51

153 143 1) k K Θ v (sk) =ˇδ(k) 1 Θ v (s), 2) k K Θ v (sk )=Θ δ (k )v(s) u V δ u, Θ v (sk) = ϑ U(u v) (sk) =ϑ U(δ(k)u v) (s) = δ(k)u, Θ v (s) = u, ˇδ(k) 1 Θ v (s), u, Θ v (sk ) = ϑ U(u v) (sk )=ϑ U(u δ (k )v)(s) = u, Θ δ (k )v)(s). f A δ ( Γ\ G, ρ G,π) v, F f (h) = f(g), Θ v (gh) d G(g) Γ\ G v V δ,h H F f : H V δ V δ {u 1,,u d } f i (g) =(f(g),u i )(g G) f i L 2 ( Γ\ G, ρ 1 G ;ˇπ, ˇδ) v V δ v, F f (h) = d F fi,ϕ i (h) (ϕ i = U(u i v) S(W )) i=1 1) γ Γ F f (γ h)=ρ H (γ )F f (h), 2) k K F f (hk )=ˇδ (k ) 1 F f (h), 3) ψ C c ( H, ˇδ ) o F f (hy 1 )ψ(y)d H(y) = ψˇπ,ˇδ (ψ) F f (h) H F f Γ \ H F f Aˇδ ( Γ \ H,ρ 1 H, ˇπ ) 6.3 G G = K G π = δ Γ ={1} L 2 ( Γ\ G, ρ 1 G ;ˇπ, ˇδ) G L 2 ( G) ˇδ- L 2 ( G; ˇδ) V δ C V δ L 2 ( G; ˇδ) (v α [x (dim δ) 1/2 v, ˇδ(x)α ]) 52

154 144 f A δ ( Γ\ G, ρ G,π) f(x) =δ(x) 1 f(1) ( k K = G) A δ ( Γ\ G, ρ G,π) V δ (f f(1)) f A δ ( Γ\ G, ρ G,π) v, F f (h) = f(1), Θ v (h) v V δ III Jacobi (V,D) 3.2 Jacobi Sp(V ) J Lie Lie sp(v ) J = sp(v ) V R Lie [(X, x, s), (Y,y,t)] = ([X, Y ],xy yx, D(x, y)) exp : sp(v ) J Sp(V ) J exp(x, x, s) =(expx, x e(x),s+2 1 D(x f(x),x)) e(x) = n=0 X n (n +1)!, f(x) = X n (n +2)! I J = I V g =(σ, h) Sp(V ) J (h =(x, t) H(V )) Z =(I,v) I J n=0 g(i,(v, 0))g 1 =(σiσ 1, (v + xi x)σ 1,D(x v vi, xi)) (g, Z) g Z =(σiσ 1, (v+xi x)σ 1 ) Sp(V ) J I J (I,0) I J U(V I,, I ) Z(H(V )) I 0 I K = U(V I0,, I0 ) p ± J = p± W ± sp(v C ) J Lie K J = K Z(H(V )) K J,C = K C Z(H(V C )) Sp(V ) J Sp(V C ) J P ± J =expp± J = P ± W ± Sp(V C ) J exp(x, x, 0) = (exp X, x, 0) p ± J P ± J 53

155 145 P + J K J,CP J = P + K C P H(V C ) Sp(V C ) J (p, k, q) pkq P + J K J,C P J P + J K J,CP J Sp(V ) J P + J K J,CP J, K J = Sp(V ) J K J,C P J Sp(V ) J /K J Sp(V ) J K J,C P J /K J,CP J P + J K J,CP J P + J log p + J I J Sp(V ) J /K J p + J D J 1) D J p + J 2) g Sp(V ) J Z D J p + J g exp Z =exp(g(z)) J(g, Z) q g(z) D J, J(g, Z) K J,C q P J 3) (g, Z) g(z) Sp(V ) J D J p + Sym C (W,W + ) D J Sym C (W,W + ) W + 1) D J = D W +, 2) g =(σ, h) Sp(V ) J (h =(x, s) H(V )) Z =(z,w) D J g(z) =(σ(z), (w+x z+z + )J(σ, z) 1 ) (x =(x,x + ) V C = W W + ), J(g, Z) =(J(σ, z), 0,η) K J,C η = s D(x, (x z + x + )J(σ, z) 1 σ) + D(x, wj(σ, z) 1 σ)+ 1 2 D(w, wj(σ, z) 1 σ) (V,D) V = W W W ρ = W C,W + ρ = W C ρ Sp(V C ) p ± J =Ad(ρ 1 )p ± J, P ± J =exp p ± J = P ± W C KJ,C = ρ 1 K J,C ρ = K C Z(H(V C )) P + J K J,C P J + Sp(V C ) J (p, k, q) pkq P J K J,C P J + P K J J,C P J ρsp(v ) J P + J K J,CP J Z D J p + J ρ exp Z exp ρ(z)k J,CP + J ρ(z) p+ J Ẑ =Ad(ρ 1 )ρ(z) p + J p+ =Sym C (W C,W C ) {Ẑ p + J Z D J} = H V W C g Sp(V ) J Z D J g(ẑ) =ĝ(z) Sp(V ) J H V,J = H V W C 54

156 146 g =(σ, h) Sp(V ) J (σ Sp(V ),h =(x, s) H(V )) Z =(z,w) H V,J (z H,w W C ) g(z) =(σ(z), (w + x z + x )J(σ, z) 1 ) (x =(x,x ) V = W W ) Z p + J exp Z Sp(V ) J K J,C P J g exp Z =expg(z) J(g, Z) q J(g, Z) K J,C,q P J J(g, Z) =(J(σ, z),η) η = s D(x, (x z + x )J(σ, z) 1 σ) + D(x, wj(σ, z) 1 σ)+ 1 2 D(w, wj(σ, z) 1 σ). Ǩ J,C = ρ 1 K J,C ρ = ǨC Z(H(V C )) (ǨC = ρ 1 K C ρ) Z, Z H V,J exp Z 1 exp Z P K(Z,Z) 1 P K(Z,Z) ǨJ,C Z =(z,w),z =(z,w ) (z,z H V,w,w W C ) K(Z,Z)=(K(z,z),κ) [ ] 1 K(z 0 z z,z)= (z z) 1, κ = (w w)(z z) 1,w w 7.2 { GSp(V )= σ GL R (V ) D(xσ, yσ) =ν(σ)d(x, y) for x, y V, ν(σ) R } Sp(V ) σ GSp(V ) X sp(v ) exp(t Ad(σ)X) =σ exp(t X)σ 1 ( t R) Ad(σ)X = σxσ 1 σ GSp(V ) h =(x, t) H(V ) h σ =(xσ, ν(σ)t) H(V ) GSp(V ) J = GSp(V ) H(V ) Sp(V ) J σ GSp(V ) (X, x, s) sp(v ) J exp(t Ad(σ)(X, x, s)) = σ exp(t(x, x, s))σ 1 ( t R) 55

157 147 GSp(V ) sp(v ) J Ad(σ)(X, x, s) =(σxσ 1,xσ 1,ν(σ) 1 s) GSp + (V )={σ GSp(V ) ν(σ) > 0} [ ν td ] 1 0 τ = GSp + (V ) (0 <ν= ν(τ) R,d GL R (W )) 0 d Z =(z, w) H J H J p + J sp J(V C ) [ ] 0 z Z =(, (0,w), 0) sp(v C ) V C C 0 0 Ad(τ)Z =(ν td 1 zd 1,wd 1 ) H V,J Ad(τ)z = ν td 1 zd 1 H V g =(σ, h) Sp(V ) J g exp Z =expg(z) J(g, Z) q (J(g, Z) =(J(σ, z),η)) τgτ 1 exp(ad(τ)z) =τg exp Zτ 1 =exp(ad(τ)g(z)) τj(g, Z)τ 1 τqτ 1, (τgτ 1 )(Ad(τ)Z) = Ad(τ)(g(Z)) τj(g, Z)τ 1 =(τj(σ, z)τ 1,ν 1 η) J(τστ 1,z)=τJ(σ, z)τ 1 η(τgτ 1 ;Ad(τ)Z) =η(g; Z) ν(τ) 1, η(τ 1 gτ; Z) =η(g;ad(τ)z) ν(τ) (5) Λ V Q D(Λ, Λ) Z Z- χ(0,t)=e(t) χ : H(Λ) C 1 Sp(Λ; χ) ={γ Sp(Λ) χ(h γ )=χ(h) for h H(Λ)} 12 η(g; Z) =e(η) η(g; Z) ν(τ) = e(ν(τ) η) 56

158 148 α(x) =χ(x, 0) (x Λ) D x α(x) 2 Ker α 2 Λ 0 <N Z Sp(Λ,N)={γ Sp(Λ) xγ x (mod N Λ) for x Λ} Sp(Λ; χ) Sp(Λ,N) Γ Sp(Λ; χ) Γ J =Γ H(Λ) g =(γ,h) Γ J χ J (g) =χ(h) (δ, V δ ) KC = GL C (W C ) Young m 1 m 2. m n m n > 0 F : H V,J V δ (H V,J = H V W C ) g =(γ,h) Γ J F (g(z)) = χ J (g)η(g; Z) 1 J δ (γ,z)f (Z) (Z =(z,w) H V,J ) Z- L W Q,L W Q ( ) 1 a) L L Λ (u, v) L L α(u+v) =e u, v, 2 {[ ] } 1 b b) 0 1 b Sym Z(L,L), L = {x W x, L Z} Sym Z (L,L)={b Sym R (W,W) L b L} 1) b Sym Z (L,L) F (z + b, w) =F (z,w), 2) l L,l L F (z, w + l z + l) =J Qz,α(l z + l, w)f (z,w). (6) l L F (z,w + l) =F (z,w) F F (z, w) = T Sym Z (L,L) Fourier λ L a(t,λ)e(tr(tz)+ l, w ) Sym Z(L,L)={T Sym R (W, W ) tr(tb) Z for b Sym Z (L,L)} 57

159 149 h =((a, b),t) H(V Q )(a W Q,b W Q ) F h (z) =η(h; z,0)f (h(z,0)) = e (t + 12 ) a, az + b F (z,az + b) (z H V ) σ Sp(V ) hσh 1 = σh σ h 1 =((a, b)(σ 1), 1 D((a, b)σ, (a, b))) 2 0 <N Z γ Sp(Λ,N) Γ hγh 1 Γ J =Γ H(Λ) F h (γ(z)) = η(h; γ(z, 0))F (hγh 1 h(z,0)) = η(h; γ(z, 0))η(hγh 1 ; h(z,0)) 1 J δ (γ,z)f (h(z,0)) = η(γ; z,0) 1 η(h 1 ; h(z,0)) 1 J δ (γ,z)f (h(z,0)) = J δ (γ,z)f h (z) F (z, w) Fourier F h (z) = a(t,λ)e (tr(tz)+ λ, az + b + 12 ) a, ax + b + t T,λ = ( ( a(t,λ)e tr T 1 t λλ + 1 t ) ) (λ + a)(λ + a) z + λ, b 2 2 T,λ ( ) 1 e a, b + t 2 a W t aa Sym R (W, W ) tr( t aa)s = a, as for S Sym R (W,W) 1) a(t,λ) 0 T 1 t λλ 0 2 2) h H(V Q ) 0 < y 0 Sym R (W,W) F h (z) {z H V Im z y 0 } Z- L,L a), b) c) L = {y W L,y Z} L 2L (6) z H V F (z, w) = v L /L F (z, ) L(Q z, 1) f v (z)ϑ[v, 0](z,w) 58

160 150 f v (z) z H V b Sym Z (L,L) l L,v L l + v, lb = l, (l + v)b L,L b L,L L, 2L az, l, vb L,L b 2Z, v, vb L,L b L,L Z l + v, (l + v)b v, vb (mod 2Z) ( ) 1 ϑ[v, 0](z + b, w) =e v, vb ϑ[v, 0](z,w) 2 F (z + b, w) =F (z,w) f v (z + b) =e ( 12 ) v, vb f v (z) (v L ) ( ) 1 fv (z) =e v, vz f v (z) b Sym Z (L,L) 2 fv (z + b) = f v (z) f v (z) f v (z) = a v (T )e (tr(tz) 12 ) v, vz F (z,w) = = T Sym Z (L,L) T Sym Z (L,L) ( a v (T )e tr(t 1 ) t vv)z 2 f v (z)ϑ[v, 0](z,w) v L /L = ( ) 1 f v (z) e l + v, (l + v)z + l + v, w 2 v L /L l L = ( ) 1 f λ(z)e λ, λz + λ, w 2 λ L = a λ (T )e(tr(tz)+ λ, w ) λ L T Sym Z (L,L) a(t,λ)=a λ (T )(T Sym Z(L,L),λ L ) Jacobi Jacobi V δ H V,J = H V W C F 1) g =(γ,h) Γ J =Γ H(Λ) F (g(z)) = χ J (g)η(g; Z) 1 J δ (γ,z)f (Z) (Z =(z,w) H V,J ) 59

161 151 2) h H(V Q ) F h (z) =η(h;(z,0))f (h(z,0)) (z H V ) 3.3 δ Siegel F Γ J χ δ Jacobi 3) h H(V Q ) F h (z) (z H V ) 3.3 Siegel F Jacobi Γ J χ δ Jacobi Koecher dim R V>2 2) 1) Γ J χ δ Jacobi S δ (Γ J,χ) F Jacobi Γ J \H V,J (δ((im 0) 1 Im z)f (z,w),f(z,w)) δ κ z (w, w)d HV (z)d z (w) < (, ) δ k J δ (k, z 0 ) K V δ Hermite S δ (Γ J ) Hermite m n >n Jacobi F Γ J \H V,J (δ((im 0) 1 Im z)f (z,w),f(z,w)) δ κ z (w, w)d HV (z)d z (w) < F Jacobi 7.4 Sp(V ) V = Sp(V ) H(V ) H V,J = H V W C (z 0, 0) H V,J K Z(H(V )) K Z(H(V )) V δ (k, t) J δ (k, z 0 ) e( t) δ e Ind Sp(V ) J K Z(H(V )) (δ e) H V,J H V,J (δ((im 0) 1 Im z)ϕ(z,w),ϕ(z,w)) δ κ z (w, w)d HV,J (z,w) < ϕ : H V,J V δ E δ e (ϕ, ψ) = (δ((im 0) 1 Im z)ϕ(z,w),ψ(z,w)) δ κ z (w, w)d HV,J (z,w) H V,J Hilbert d HV,J (z,w) =d HV (z)d z (w) H V,J Sp(V ) J - J δ e (g, Z) =J δ (σ, z)η(g; Z) 1 (g =(σ, h) Sp(V ) J,Z =(z,w) H V,J 60

162 152 E δ e Sp(V ) J π δ e (π δ e (g)ϕ)(z) =J δ e (g 1,Z)ϕ(g 1 (Z)) Ind Sp(V ) J K Z(H(V )) (δ e) (π δ e,e δ e ) ϕ E δ e H δ e E δ e Sp(V ) J - H δ e Sp(V ) J - Sp(V ) J (π δ e,h δ e ) Sp(V )J = Sp(V ) H(V ) 5.4 Sp(V ) Sp(V ) J ω χ,j ϕ H δ det 1/2 ψ H z0 H V,J ϕ ψ (ϕ ψ)(z, w) = det(im z) 1/4 ϕ(z) (U z,z0 ψ)(w) (z H V,w W C ) ϕ ψ ϕ ψ Sp(V )J (π δ det 1/2,H δ det 1/2) (ˇω χ,j, H z0 ) (π δ e,h δ e ) (7) π δ det 1/2 Sp(V )J Sp(V ) Sp(V )J H δ e {0} m n >n (π δ e,h δ e ) Sp(V )J Jacobi F S δ (Γ J,χ) Sp(V ) J f F f F (g) =J δ e (g;(z 0, 0)) 1 F (g(z 0, 0)) (g Sp(V ) J ) 1) g Γ J f F (g g)=χ J (g )f F (g), 2) k K f F (gk) =δ(k) 1 f F (g), 3) f F (g) 2 d(ġ) Γ J \Sp(V ) J = (δ((im 0) 1 Im z)f (z,w),f(z,w)) δ κ z (w, w)d HV (z)d z (w) < Γ J \H V,J F f F S δ (Γ J,χ) A δ (Γ J \Sp(V ) J,χ 1 J,π δ e) F S δ (Γ J,χ) α V δ θ F α (g) =(dimδ) 1/2 α, f F (g) (g Sp(V ) J ) 61

163 153 F α θ F α Hilbert S δ (Γ J,χ) C V δ L 2 (Γ J \Sp(V ) J ; χ J ;ˇπ δ e, ˇδ) 7.5 (V,D) V = W W Z- L W,L W L = {x W x, L Z} Λ=L L H(Λ) χ Λ : H(Λ) C 1 (((x, y),t) e (t + 12 ) x, y ) Sp 0 (Λ) = {γ Sp(Λ) χ λ (h γ )=χ Λ (h) for h H(Λ)} Sp(Λ) 0 <N Z Sp(Λ,N) Sp 0 (Λ) Sp(Λ,N) Γ Sp 0 (Λ) Γ Γ J =Γ H(Λ) χ Λ,J :Γ J C 1 ((γ,h) χ Λ (h)) π χ Λ,J =Ind Sp(V ) J Γ J χ Λ,J Sp(V ) ϖ : Sp(V )= Sp(V ; z 0 ) Sp(V ) Γ ϖ 1 (Γ) Sp(V ) Sp(V )J = Sp(V ) H(V ) ϖ J : Sp(V ) J SP(V ) J π χ Λ,J = π χ Λ,J ϖ J Sp(V )J π χ Λ,J =Ind Sp(V ) J Γ J χ Λ,J ( Γ J = Γ H(Λ), χ Λ,J = χ Λ,J ϖ J ) π χ Λ,J Sp(V ) Weil Ind H(V ) H(Λ) χ ) Λ ϕ Ind Sp(V ρ 1 Γ Λ ψ IndH(V ) H(Λ) χ Λ (ϕ ψ)( g) =ϕ( σ)(ω J ( g)ψ)(1) ( g =( σ, h) Sp(V ) J ) ( γ,h ) Γ J (ϕ ψ)(( γ,h ) g) =ϕ( γ σ)(ω J ( γ,h ) ω J ( g)ψ)(1) ϕ ψ Ind Sp(V ) J Γ J = ρ Λ ( γ) 1 ϕ( σ)(ρ Λ ( γ)r(γ) Π(h ) ω J ( g)ψ)(1) = ϕ( σ)(ω J ( g)ψ)(h )=χ Λ (h )ϕ( σ)(ω J ( g)ψ)(1) = χ Λ,J ( γ,h )(ϕ ψ)( g) χ Λ,J 62

164 ϕ ψ ϕ ψ Sp(V )J ) (Ind Sp(V ρ 1 Γ Λ ) ) J ω J J Ind Sp(V χ Γ Λ,J J GL C (W C ) (δ, V δ ) Young m 1 m 2. m n m n >n Sp(V )J (7) ˇπ δ det 1/2 ω J ˇπ δ e ˇπ δ e K- ˇδ ˇπδ det 1/2 K- ˇδ det 1/2 ω J K- det 1/2 5.3 ψ Ind H(V ) H(Λ) χ Λ det 1/2 - (ω J ( g)ψ)(1) = ε η(g; z 0, 0) det(2imσ(z 0 )) 1/4 ϑ(g(z 0, 0)), ( g =( σ, h) Sp(V ) J ) ϑ(z, w) = ( ) 1 e l, lz + l, w 2 l L ((z,w) H V,J = H V W C ) Jacobi F S δ (Γ J,χ Λ ) α Vδ Ind Sp(V ) J Γ J χ J θ F α θ F α (g) =θ f α ( σ) (ω J ( g)ψ)(1) (g =(σ, h) Sp(V ) J ) f S δ det 1/2( Γ,ρ 1 Λ ) η(g; z 0, 0)J δ (σ, z 0 ) 1 F (g(z 0, 0)) =J 1/2 ( σ, z 0 )J δ (σ, z 0 ) 1 f(σ(z 0 )) ε η(g; z 0, 0) det(2imσ(z 0 )) 1/4 ϑ(g(z 0, 0)) ( σ =(ε, σ) Sp(V )) F (z,w) = det(2im z 0 ) 1/4 f(z)ϑ(z,w) ((z,w) H V,J )

165 m n >n Jacobi F S δ (Γ J,χ Λ ) f(z) = det(2im z 0 ) 1/4 det(2im z) 1/2 F (z,w)ϑ(z,w)κ z (w, w)d z (w)(z H V ) W C /Λ z F f S δ (Γ J,χ Λ ) S δ det 1/2( Γ,ρ Λ ) [ ] 7.6 V = R 2n D(x, y) =xj t 0 1 n n y (J n = ) 1 n 0 (V,D) V = W W W = {(x, 0) x R n }, W = {(0,y) y R n } (x, 0) = x, (0,y)=y W = R n, W = R n W,W Z- L = {(x, 0) x Z n }, L = {(0,y) y Z n } Λ=L L f S k 1/2 (Γ 0 (4)) [ ] f (z) =f(z/2) = f(ad(τ )z) (τ = GSp(V )) 0 1 f S δ det 1/2(Γ,ρ Λ ) {[ ] } Γ=τΓ 0 (4)τ 1 a b = Sp(Λ) c d b c 0 (mod 2) Sp 0 (Λ) GL C (W C ) δ =det k F J cusp k,1 F (z,w) =F (2z,w) =F (Ad(τ)(z,w)) g =(γ,h) τ 1 Sp(Λ) J τ = τ 1 Sp(Λ)τ H(Λτ) F (g(z, w)) = χ Λ (h)η(g; z,w) 1 det J(γ,z) k F (z,w) F (z, w) = h H(Λτ)\H(Λ) χ Λ (h) 1 η(h; z,w)f (h(z,w)) 64

166 156 F S δ (Γ J,χ Λ ) F S δ det 1/2(Γ,ρ Λ ) r, s L W C /Λ z θ r (2z,w)θ s (2z,w)κ z (w, w)d z (w) 2 n det(2im z) 1/2 r s (mod 2L ), = 0 r s (mod 2L ) (Λ z = {xz + y x 2L,y L}) ϑ(z,w) = r L /2L θ r (2z,w) F (z,w)ϑ(z,w)κ z (w, w)d z (w) W C /Λ z = F (z, w + λ)ϑ(z,w + λ)κ z (w + λ, w + λ)d z (w) = W C /Λ z λ Λ z /Λ z W C /Λ z F (z,w)ϑ(z,w)κ z (w, w)d z (w) =2 n det(2im z) 1/2 r L /2L f r (2z) F S δ (Γ J,χ Λ ) f r (2z) S δ det 1/2(Γ,ρ Λ ) r L /2L f r (4z) S k 1/2 (Γ 0 (4)) r L /2L Tamagawa [18], Gaal [4] 2.1 G K A G Z(G) χ A K 1 K H = C c (G/A, χ; 1 K ) H G ϕ 1) a A ϕ(ax) =χ(a) 1 ϕ(x), 2) G/A ẋ ϕ(x) 3) k, k K ϕ(kxk )=ϕ(x) 65

167 157 G/A (ϕ ψ)(x) = ϕ(xy)ψ(y 1 )d G/A (ẏ) G/A C- θ G a A θ(ax) =χ(a)θ(x) ϕ H θ(ϕ) = ϕ(x)θ(x)d G/A (ẋ), ˇθ(ϕ) = ϕ(x)θ(x 1 )d G/A (ẋ) G/A G/A G G θ 1) a A θ(ax) =χ(a)θ(x), 2) k K θ(kxk 1 )=θ(x), 3) θ : H C C- θ χ K G Θ(G/A, χ, K) θ Θ(G/A, χ, K) 1) ϕ H ϕ θ = θ ϕ = ˇθ(ϕ) θ, 2) θ K- θ(1) = 1, 3) θ(xky)d K (k) =θ(x)θ(y). K G θ a A θ(ax) = χ(a)θ(x) G/A ẋ θ(x) G/A 1) θ Θ(G/A, χ, K), 2) θ K- θ(1) = 1 ϕ H ϕ θ = λ ϕ θ (λ ϕ C), 3) θ 0 θ(xky)d K (k) =θ(x)θ(y), K 4) k K θ(kxk 1 )=θ(x) θ : H C C-. H M G/A supp[ẋ ϕ(x) ] M ϕ H H M H M ϕ = 66

168 158 sup ϕ(x) Banch H = H M ẋ G/A M V H H M G/A V H M H M 0 <r R {ϕ H ϕ <r} H H Hausdorff ) θ Θ(G/A, χ, K) C- θ : H C 2) C- λ : H C θ = λ θ Θ(G/A, χ, K) Hecke K G K G H C C- θ θ Θ(G/A, χ, K) Hom C-alg (H, C) \{0} 8.2 p 2 Q p χ = e p e p : Q p Q p /Z p Q/Z ṫ e 2π 1t C 1 Q p - (V,D) V = W W Z p - L W L = {x W x, L Z p } ( x, y = D(x, y) forx W,y W ) Z p - Λ=L L V p 2 H[Λ]=Λ Z p H(V ) K = {σ Sp(V ) Λσ =Λ} Sp(V ) H[Λ] (x, t) σ =(xσ, t) K J = K H[Λ] Sp(V ) J K J 1 1 KJ 8.1 H J = C c (Sp(V ) J /Z(H(V )),χ; 1 KJ ) H J Sp(V ) J ϕ 1) a Z(H(V )) = Q p ϕ(ag) =χ(a) 1 ϕ(g), 67

169 159 2) k, k K J ϕ(kgk )=ϕ(g), 3) Sp(V ) J /Z(H(V )) ġ ϕ(g) (ϕ ψ)(g) = ϕ(gx 1 )ψ(x)d Sp(V )J /Z(H(V ))(ġ) Sp(V ) J /Z(H(V )) C- L Z p - {v 1,,v n } v i L v i,v j = δ ij α =(α 1,,α n ) Z n p α GL Qp (W )s.t.p α v i = p α i v i, p α GL Qp (W )s.t.p α v i = p α i v i [ ] t p α = p α d(p α p α 0 )= 0 p α Sp(V ) Sp(V )= α Υ Kd(p α )K (Υ = {(α 1,,α n ) Z n α 1 α n 0}) ϕ H J suppϕ α Υ K J d(p α )K J Z(H(V )) α Υ Sp(V ) J K J d(p α )K J ϕ α ϕ α C c (Sp(V ) J ; 1 KJ ) o 7 ϕ [ α,χ H J ] {ϕ α,χ } α Υ H J C- 0 1 n ε = GSp(n, Q p )=GSp(V ) 1 n 0 g g = εg 1 ε 1 Sp(V ) J ϕ α,χ (g )=ϕ α,χ (g) (α Υ) H J Murase [13] H J H J C 8.3 p 2 k K r χ(k) =(k, r χ (k)) Sp(V ) K = r χ (K) Sp(V ) p α GL Qp (W )(α Z n ) (3) d(p α ) Sp(V ) Sp(V )= K d(p α ) KKer(ϖ χ ) α Υ α Υ Sp(V ) K d(p α ) K ψ α ψ α C c ( Sp(V ); 1 K) o 7 ψ α,ν H= C c ( Sp(V )/Ker(ϖ χ ),ν; 1 K) o ν 68

170 160 Ker(ϖ χ ) {ψ α,ν } α Υ H C (ω J,L 2 (W )) Sp(V )J K- L ϕ L Sp(V )J Φ(g) =(ω J (g)ϕ L,ϕ L )(g Sp(V ) J ) 1) α Z n Φ( d(p α )) = (p 1/2 η p ) α, α = α α n η p = γ χ (Q 1 )γ χ ( p Q 1 ) 2) supp(φ) = K J d(p α ) K J (Ker(ϖ χ ) Z(H(V ))) α Υ Shintani [17] ϕ H ϕ J (σ, h) =ϕ( σ) Φ( σ, h) ((σ, h) Sp(V ) J, σ Sp(V )s.t.ϕ χ ( σ) =σ) ϕ ϕ J H H J C- 8.4 Sp(V ) Jacobi Sp(V )J θ Θ( SP(V )/Ker(ϖ χ ),ν, K) Sp(V ) J θ J θ J (σ, h) =θ( σ)φ( σ, h) (ϖ( σ) =σ) θ J Θ(Sp(V ) J /Z(H(V )),χ,k J ) H J H C- {ϕ α,χ } α Υ {ψ α,ν } α Υ T : H J H Tϕ α,χ =(p 1/2 ηp 1 ) α ψ α,ν (α Υ) ) T H J H C ) θ Θ( Sp(V )/Ker(ϖ),ν, K) θ T = θj θ θ J Θ( Sp(V )/Ker(ϖ χ ),ν, K) Θ(Sp(V ) J /Z(H(V ))),χ,k J ) Sp(V ) τ τ Ker(ϖ) = ν K- R( Sp(V )/Ker(ϖ),ν, K) Sp(V ) J π π Z(H(V )) = χ K J - R(Sp(V ) J /Z(H(V )),χ,k J ) H, H J 69

171 161 K- KJ τ π = τ J ω J R( Sp(V )/Ker(ϖ),ν, K) R(Sp(V ) J /Z(H(V )),χ,k J ) θ π =(θ τ ) J [1] W.L.Baily,Jr. and A.Borel : Compactification of arithmetic quotients of bounded symmetric domains (Ann. of Math. 84 (1966), ) [2] J.Dixmier : C -algebras (North-Holland, 1982) [3] M.Eichler and D.Zagier : Theory of Jacobi Forms (Progress in Math. 55 (1985)) [4] S.A.Gaal: Linear analysis and representation theory (Die Grund. math. Wiss. Einzel. 198, Springer-Verlag, 1973) [5] S.Gelbart, I.Piatetski-Shapiro, S.Rallis : Explicit Constructions of Automorphic L-Functions (Lecture Notes in Math. 1254, Springer-Verlag, 1980) [6] R.Godement : Généralités sur les formes modulaires, I,II (Séminaire H.Cartan, 1957/58) [7] Harish-Chandra : Invariant eigen distributions an a semisimple Lie group (Trans. Amer. Math. Soc. 119 (1965), ) [8] Harish-Chandra (Notes by G. van Dijk) : Harmonic Analysis on Reductive p-adic Groups (Lecture Notes in Math. 162, Springer-Verlag, 1970) [9] R.Howe : Transcending classical invarinat theory (J. of American Math. Soc. 2 (1989), ) [10] T.Ibukiyama : On Jacobi forms and Siegelmodular forms of half integral weights (Comment. Math. Univ. St. Pauli 41 (1992), [11] J.-I.Igusa : Theta Functions (Die Grundlehren der math. Wiss. Einz. 194, Springer-Verlag, 1972) 70

172 162 [12] G.Lion, M.Vergne : The Weil representation, Maslov index and Theta series (Progress in Math. vol.6, Birkhäuser, 1980) [13] A.Murase : L-functions attached to Jacobi forms of degree n, PartI. The basic identity (J. reine angew. Math. 401 (1989), ) [14] I.Satake : Caractérisation de l espace des Spitzenformen (Séminaire H. Cartan 1957/58) [15] I.Satake : Factors of automorphy and Fock representations (Advances in Math. 7 (1971), ) [16] I.Satake : Algebraic Structures of Symmetric Domains (Math. Soc. Japan, Iwanami-Shoten and Princeton Univ. Press, 1980) [17] T.Shintani : unpublished nonte [18] T.Tamagawa : On Selberg s trace formula (J. Fac. Sci. Univ. Tokyo 8 (1960), ) [19] A.Weil :Sur certauns groupes d opérateurs unitaires (Acta math. 111 (1964), ) 71

173 163 SAITO KUROKAWA LIFTING FOR LEVEL N 1. Introduction N N Saito Kurokawa lifting ( Maass lift [9] [13] [13] [9] Saito Kurokawa lift [9] [3], [4], [12], [19] [9] H n n (1) N trivial index 1 H 1 C N (2) (3) L (4) N index Kramer index 1 (cf. [9]) 2. n H n H n = {Z = t Z M n (C); Z = X + iy, X, Y M n (R), Y > 0}

174 164 n Sp(n, R) Sp(n, R) = {g M 2n (R); gj t n g = J n }, ( ) 0 1n J n = 1 1 n 0 n n GSp + (n, R) GSp + (2, R) = {g M 2n (R); gj n t g = n(g)j n, 0 < n(g) R}. GSp + k H n F (Z) GSp + (n, R) g = ( C A D B ) GSp+ (n, R) (F k [g]) = det(cz + D) k F (gz), gz = (AZ + B)(CZ + D) 1 Sp(2, R) J(R) a 0 b µ J(R) = λ 1 µ κ c 0 d λ P 1 (R), ad bc = 1 (a, b, c, d R), λ, µ, κ R 0 P 1 (R) = 0 Sp(2, R) x C e(x) = exp(2πix) Z H 2 Z = ( τ z ω z ) H 1 C f(τ, z) ω H 1 m F (Z) = f(τ, z)e(mω) g J(R) Sp(2, R) F k [g] = f(τ, z)e(mω) ω f(τ, z) f f J(R) H 1 C f 2.1 J(R) (g, ((λ, µ), κ)) (g SL 2 (R)) J(Z) = SL 2 (Z) J H(R) = {(1, ((λ, µ), κ)); λ, µ, κ R} Heisenberg group ((λ, µ), κ) κ λ, µ Z (λ, µ) = ((λ, µ), 0) J(R) Sp(2, R)

175 165 Γ Γ J(R) H(Z) = {(1, ((λ, µ), κ); λ, µ, κ Z} Γ J(R) H(Z) H(Z) J(R) H(Z) H(Z) (Saito-Kurokawa lift) {( ) } a b Γ 0 (N) = SL c d 2 (Z); c 0 mod N Γ 0 (N) J = {(g, 1); g Γ 0 (N)} H(Z) J(Z) N χ ( modulo ) N Dirichlet χ(g) = χ(d) a b g = Γ c d 0 (N) Γ 0 (N) J Γ 0 (N) J H 1 C f(τ, z) g GL 2 (R) (det(g) = l > 0) ((λ, µ), κ) H(R) ( (f k,m [g])(τ, z) = (cτ + d) k e ml cz2 cτ + d ) ( aτ + b f cτ + d, (f m [((λ, µ), κ)])(τ, z) = e m (λ 2 τ + 2λz + λµ + κ)f(τ, z + λτ + µ). ) lz, cτ + d (τ, z) H 1 C q = e(τ), ζ = e(z) H 1 C f(τ, z) k index m Γ 0 (N) J χ (i) f k,m [g] = χ(g)f (g Γ 0 (N)). (ii) f m [(λ, µ)] = f (λ, µ Z). (iii) g SL 2 (Z) f k,m [g] = c g (n, r)q n ζ r n,r Q 4nm r 2 0 c g (n, r) = 0 J k,m (Γ 0 (N) J, χ) ϕ (iii) (iii ) (iii ) g SL 2 (Z) 4nm r 2 > 0 c g (n, r) = 0. J cusp k,m (Γ 0(N) J, χ) level 1 ([5]) level N

176 166 Lemma 2.1. f J k,m (Γ 0 (N) J, χ) f f(τ, z) = c(n, r)q n ζ r n,r Z,4nm r 2 0 c(n, r) 4nm r 2 r mod 2m m = 1 4n r 2 f(τ, z + λτ + µ) = e( m(λ 2 τ + 2λz))f(τ, z) c(n, r)q n+mλ2 +rλ ζ r+2mλ = c(n, r)q n ζ r n,r n,r λ (1) c(n, r) = c(n + mλ 2 + rλ, r + 2mλ) n 1 = n+mλ 2 +rλ, r 1 = r+2mλ 4n 1 m r 2 1 = 4nm r 2, r r 1 mod 2m c(n, r) 0 (n, r) (n 1, r 1 ) r r 1 mod 2m 4n 1 m r 2 1 = 4nm r 2 r 1 = r + 2mλ λ Z 4n 1 m = r nm r 2 = (r +2mλ) 2 +4nm r 2 = 4mλ 2 +4mrλ+4nm. n 1 = n+mrλ+mλ 2. c(n 1, r 1 ) = c(n+mλ 2 +rλ, r+2λ) = c(n, r) m = 1 4n r 2 r mod 2 c(n, r) 4n r 2 q.e.d Theta expansion. m ν Z/2mZ H 1 C ϑ ν,m (τ, z) = e((p + ν 2m )2 mτ + (p + ν 2m )(2mz)) p Z (ii) H 1 C f(τ, z) C H 1 c ν (τ) (2) f(τ, z) = c ν (τ)ϑ ν (τ, z). ν Z/2mZ c ν (τ) f(τ, z) f (ii) (i) 1 2 Γ 0 (N) f(τ, z) = ( 1) k χ( 1)f(τ, z) m = 1 ϑ ν,1 (τ, z) z χ( 1) = ( 1) k f = 0 m 1 χ( 1) = ( 1) k f z ϑ ν,m (τ, z) = ϑ ν,m (τ, z) χ( 1) = ( 1) k

177 167 (2) c ν (τ) = c ν (τ) (3) f(τ, z) = c ν (τ) (ϑ ν (τ, z) + ϑ ν (τ, z))/2 ν Z/2mZ ν ν mod 2m ν 0 mod m ν = 0, ν = m ϑ ν (τ, z) = ϑ ν (τ, z) 0 ν m θ ν (τ, z) = 1 2 (ϑ ν(τ, z) + ϑ ν (τ, z)) f(τ, z) = m c ν (τ)θ ν (τ, z) ν=0 (f(τ, z) m Taylor expansion. f(τ, z) z C z = 0 τ Eichler-Zagier (cf. [10]) Saito-Kurokawa lift Eichler-Zagier Kramer [13] N Kramer f(τ, z) J k,m (Γ 0 (N) J, χ) z Taylor g SL 2 (Z) f k,m [g](τ, z) = χ l,g (τ)z 2l χ l,g (τ) H 1 g = 1 2 χ l,1 = χ l f k,m [g] (ii) H(Z) SL 2 (Z) (ii) l=0

178 168 f k,m [g] f f k,m [g] ( 1 2 SL 2 (Z) f k,m [g](τ, z) = m c ν,g (τ)θ ν (τ, z) ν=0 θ ν (τ, z) g 1 τ f(τ, z) z 2m ([5]) f χ l (τ) (l = 0,...,m) χ l (τ) l = 0,...,m f = 0 χ l,g (τ) χ l (τ) f(τ, z) Taylor χ l (τ) (4) ξ k,l (τ) = l] µ=0 (k + 2l µ 2)! ( 2πim) µ dµ µ!(k + 2l 2)! dτ χ l µ(τ) µ ξ k,l (τ) A k+2l (Γ 0 (N), χ) (Eichler-Zagier ξ 2l (τ) A k+2l (Γ 0 (N), χ) k + 2l χ, N J k,1 (Γ 0 (N) J, χ) A k (Γ 0 (N), χ) A k+2 (Γ 0 (N), χ) A k+2m (Γ 0 (N), χ) N = 1, 2, 3, 4, m = 1, χ = id ξ k,2l (τ) χ l (τ) Eichler-Zagier [5] p. 34 (12) χ l (τ) = l µ=0 (2πim) µ (k + 2l 2µ 1)! ξ (µ) k,l µ (k + 2l µ 1)!µ! (τ) (ξ k,2l (τ)) 0 l m (ξ k+2l (τ)) 0 l m c ν (τ) χ l (τ) (ii) f(τ, z) g SL 2 (Z) f(τ, z) k,m [g] = χ l,g (τ)z 2l l=0

179 169 f(τ, z) k,m [g] = m c ν,g (τ)θ ν (τ, z) ν=0 z z = 0 2m m (5) c ν,g (τ) z 2l θ m (τ, z) z=0 = (2l)!χ l,g (τ). (0 l m). ν=0 z = heat equation z 1 2 (2πi) 2 z θ m(τ, z) = 4m 2 2πi τ θ m(τ, z) z 2l l τ τ =.) τ f(τ, z) χ l,g c ν,g (τ) χ l,g (τ) A = ( l τθ ν (τ, z)) 0 l m,0 ν m η(τ) = q 1/24 n=1 (1 qn ) η(τ) (m+1)(2m+1) 0 τ i (0 i m) m + 1 H 1 (τ 0, τ 1,..., τ m ) H1 m+1 f(τ 0,..., τ m ) 1 1 D = τ0 τm..... τ m 0 τ m m ( ) a b H 1 h(τ) g = SL c d 2 (R) f k [g] = (cτ + d) k f(gτ) k (cf. e.g. [8]) g SL 2 (R) m Res τl =τ(d(( (cτ l + d) k f(gτ 0,..., gτ m )) l=0 = (Res τl =τdf(τ 0,..., τ m )) (m+1)k+m(m+1) [g] Res τl =τ τ l τ H 1 H 1 H 1 diagonal embedding k m+1 (m+1)k

180 m = m(m+1) f(τ 0,..., τ m ) = f 0 (τ 0 )f 1 (τ 1 ) f(τ m ) f 0 (τ) f m (τ) Res τl =τ (D(f 0 (τ 0 ) f m (τ m ))) = τ f 0 (τ) τ f m (τ) τ m f 0 (τ) τ m f m (τ) f ν (τ) = θ ν (τ, 0) (0 ν m) Lemma 2.2. Res τl =τ(d(θ 0 (τ) θ m (τ)) η(τ) (m+1)(2m+1) 0 SL 2 (Z) ( ) 1 1 SL 2 (Z) T = J = 0 1 ( ) 0 1 T θ 1 0 ν (τ + 1) = e(ν 2 /4m)θ ν (τ) θ 0 (τ +1) θ m (τ +1) = e((m+1)(2m+1)/24)θ 0 (τ) θ m (τ). J Igusa [11] II ϑ ν ( τ 1 ) = κ(j) τ/2m e( rµ/2m)ϑ r (τ) r Z/2mZ κ(j) τ ϑ ν (τ) (theta functions of the second kind) Igusa θ ν ( τ 1 )τ 1/2 θ r (τ) C θν( τ 1 )(τ) 1/2 1 e( rν/2m) + e(rν/2m) = κ(j) ϑ r (τ) 2m 2 r Z/2mZ ) m 1 1 = κ(j) (θ 0 (τ) + e(ν/2)θ m (τ) + (e( rµ/2m) + e(rµ/2m))θ r (τ) 2m r=1 (θ 0 ( τ 1 ),..., θ m ( τ 1 ))τ 1/2 = (θ 0 (τ),..., θ m (τ))a τ 1/2 A 2 = α1 m+1, α = 1 Res(D(θ 0 (τ 0 )τ 1/2 0 θ m (τ m )τ 1/2 m ) = det(a)res(d(θ 0 (τ 0 ) θ m (τ m ))

181 171 Res(D(θ 0 (τ 0 ) θ m (τ m )) (m+1)/2+m(m+1) [J] f(τ) := det( d2ν θ dτ 2ν r ) SL 2 (Z) g SL 2 (Z) f (m+1)(2m+1)/2 [g] = c(g)f (c(g) η(τ) (m+1)(2m+1) SL 2 (Z) multiplier f(τ)/η(τ) (m+1)(2m+1) i f(τ) d2l θ dτ 2l ν ν 0 i q ν2 /4m f(τ) m ν=1 (ν2 /4m) = (m + 1)(2m + 1)/24 η(τ) (m+1)(2m+1) f(τ)/η(τ) (m+1)(2m+1) i Vandermonde q (m+1)(2m+1)/24 Vandermonde Lemma 2.2 (5) c ν (τ) χ l (τ) (0 l m) f(τ, z) (2m f = 0 (H n C n ( [10] ) ( index). (5) g = 1 2 χ l (τ) (0 l m) (5) g = 1 2 c l (τ) f(τ, z) = m c ν (τ)θ ν (τ, z) ν=0 f(τ, z) χ l (τ) f(τ, z) (ii) f(τ, z) (i) f k,m [g] (ii) g SL 2 (Z) gh(z) = H(Z)g

182 172 f k,m [g] = m c ν,g (τ)θ ν (τ, z) ν=0 H 1 c ν,g (τ) c ν,g (τ) c ν k 1/2 [g] z = 0 Taylor k,m [g] c ν,g (τ) Taylor ( l=0 χ l(τ)z 2l) k,m [g] Taylor (4) χ l (τ) ξ k,l (τ) f(τ, z) A k+2l (Γ 0 (N), χ) ξ k,l (τ) A k+2l (Γ 0 (N), χ) χ l (τ), (4) ξ k,l (τ) g SL 2 (Z) ξ k,l,g (τ) = ξ k,l (τ) k+2l [g] ξ k,l,g (τ) χ l,g (τ) f k,m [g] Taylor l=0 χ l,g(τ)z 2l ξ k,l f(τ, z)e(mω) H 2 H 1 H 1 H 2 (g, ((0, 0), 0)) J(R) Sp(2, R) f k,m [g] Taylor g SL 2 (Z) (5) c ν,g (τ) χ l (τ) ξ k,l (τ) A k+2l (Γ 0 ) g Γ 0 (N) ξ k,l k [g] = χ(g)ξ k,l c ν,g (τ) = χ(g)c ν (τ) f k,m [g] = χ(g)f f Γ 0 (N) ξ k,l A k+2l (Γ 0 (N), χ) f(τ, z) ( H 1 ) Koecher (Ziegler [23]) (iii) ( ) g SL 2 (Z) f k,m [g] c g (n, r) 4mn r 2 0 g Γ 0 (N)\SL 2 (Z)/P 0 (P 0 SL 2 (Z) Γ 0 (N) g SL 2 (Z) g 1 Γ 0 (N)g P 0 = { ± ( 1 ng Z 0 1 )}

183 173 0 < n g Z f k,m [g] = c g (n, r)q n/ng ζ r n,r Z 4nm r 2 0 c g (n, r) = 0 θ ν (τ, z) (0 ν m) q mp2 ±pν+ν 2 /4m ζ 2mp±ν c g (n, r) r ±ν mod 2m c ν,g (τ)θ ν (τ, z) f k,m [g] τ τ + n g c ν (τ) = α Q a αq α α + ν 2 /4m n 1 g Z α = n n g ν2 4m (1) c g (n, r) c g (α + mp 2 ± pν + ν 2 /4m, 2mp ± ν) = c g (n/n g, ±ν) 4nm/n g ν 2 0 c ν,g (τ) = n Z a ν,g (n)q n/ng ν2 /4m 4n/n g ν 2 /m a ν,g (n) = 0 (c ν,g (τ)) ν g c ν,g (τ) c ν k 1/2 [g] (c ν (τ)) multiplier χ l c ν,g (τ) χ l χ l,g ξ k,l n < 0 a ν,g (n) = 0 χ l,g (τ) ξ k,µ k [g] q c ν,g (τ) q (m + 1)(2m + 1)/24 c ν,g (τ) i 0 ν m a ν,g (n) 0 n n 0 m ν 2l a ν,g (n 0 ) (0 l m) ν=0 Vandermonde c ν,g (τ) q

184 174 n 0 < 0 a ν,g (n) = 0 if n < 0 0 n/n g < ν 2 /4m n ν n = 0 a 0,g (0) 0 < ν a ν,g (0) = 0 a ν,g (n) a ν,g (0) = 0 (0 < ν) n = 1 n = 0 1 < n g ν 2 /4m ν a ν,g (1) = 0 n/n g ν 2 /4m closed formula index (index1). m = 1 m = 1 ν = 0 ν = 1 θ ν (τ, z) = ϑ ν (τ, z) ξ k,0 (τ), ξ k,1 (τ) f 0 (τ) A k (Γ 0 (N), χ), f 1 (τ) A k+2 (Γ 0 (N), χ) f l k+2l [g] = n=0 b ν,g(n)q n/n g (5) c 0,g (τ)ϑ 0 (τ) + c 1,g (τ)ϑ 1 (τ) = (f 0 k [g])(τ), c 0,g (τ)ϑ 0(τ) + c 1,g (τ)ϑ 1(τ) = 1 2k (f 0 k [g]) (τ) + πi 4k (f 1 k+2 [g])(τ). τ ν = 1 a 1,g (n) = 0 (n < n g /4) ϑ 0 (τ, z) = 1 + q(ζ 2 + ζ 2 ) + q 4 (ζ 4 + ζ 4 ) + ϑ 1 (τ, z) = q 1/4 (ζ + ζ 1 ) + q 9/4 (ζ 3 + ζ 3 ) + f k,m [g] q n/n g (0 n/n g < 1/4) 1/4 < 1 ϑ ν a ν,g (n) q n/ng+1 (0, 1/4)

185 175 a 0,g (n) + 2a 1,g (n) = b 0,g (n) (2πi) 2 a 1,g (n) = 2πin 2kn g b 0,g (n) + πi 4k b 1,g(n) f(τ, z), a 1,n (n) = 0 (0 n < n g /4) f 0,g, f 1,g b 1,g (n) = 4n n g b 0,g (n) (0 n < n g /4) (cf. [13]) n = 0 b 1,g (0) = 0 g SL 2 (Z) f 1 S k+2 (Γ 0 (N), χ) n g 4 J k,1 (Γ 0 (N) J, χ) A k (Γ 0 (N), χ) S k+2 (Γ 0 (N), χ) ϕ (f 0, f 1 ) J k,1 (Γ 0 (N) J, χ) N = 1 Eichler-Zagier [5] N N = 5 index 1 [9] 3. Saito Kurokawa lift Saito Kurokawa lift χ ( modn Dirichlet character Γ (2) 0 (N) = { g = ( ) A B C D } Sp(2, Z); C 0 mod N χ(g) = χ(det(d)) χ Γ (2) 0 (N) 3.1. Cusp forms. ϕ(τ, z) J cusp k,1 (Γ 0(N) J, χ) Index 1 Index index 0 index V l { ( ) } a b N,0 (l) = g = ; a, b, c, d Z, det(g) = l, (a, N) = 1, cn d N,0 = N,0 (l). l=1 N,0 N,0 g = ( a c d b ) χ χ(g) = χ(a) 1 g Γ 0 (N) χ(a) 1 = χ(d)

186 176 J k,m (Γ 0 (N) J, χ) V l,χ (ϕ k,m V l,χ )(τ, z) = l k 1 g Γ 0 (N)\ N,0 (l) = l k 1 a c b d Γ 0 (N)\ N,0 (l) χ(g) 1 ϕ k,m [g] ( χ(a)(cτ + d) k e lm cz2 cτ + d ) ( aτ + b ϕ cτ + d, J k,ml (Γ 0 (N) J, χ) ( B. Ramakrishnan ϕ(τ, z) J k,1 (Γ 0 (N) J, χ) ) lz. cτ + d (L N,χ ϕ)(z) = (ϕ k,1 V l,χ )(τ, z)e l (ω). l=1 χ( 1) = ( 1) k ϕ = 0 χ trivial k Saito-Kurokawa lift ϕ(τ, z) = c(4n r 2 )q n ζ r n,r,4n r 2 >0 ( ) τ z Z = H z ω 2 (L N,χ ϕ)(z) = m=1 n,r Z,4mn r 2 >0 a (n,m,r),(a,n)=1 ( ) 4nm r χ(a)a k 1 2 c q n ζ r e(mω) Theorem 3.1. L N,χ J cusp k,1 (Γ 0(N) J, χ) S k (Γ (2) 0 (N), χ) R = a 2

187 177 ( ) a b g = SL c d 2 (R), x R, S = t S M 2 (R) a 0 b 0 ( ) u(x) = x x, ι(g) = c 0 d 0, u(s) = 12 S Γ (2) 0 (N) Lemma 3.2 (Aoki-Ibukiyama [2] p. 265 Lemma 6.2). N Γ (2) 0 (N) R, u(x), u(s) ι(m). x, S, M x Z, S = t S M 2 (Z), M Γ 0 (N) L N,χ (ϕ) R (χ Sp(2, Q) P 1 0 P 1 (Q) = 0 Sp(2, Q) Lemma 3.3. Γ (2) 0 (N)\Sp(2, Q)/P 1 (Q) P 1 (Q)R R A k (Γ 0 (N), χ) F index 0 F i1 2 F (Z) = T a(t )e(t r(t Z)) a(t ) T > non-cusp forms. ϕ c(0) 4nm r 2 = 0 q n ζ r e(mω) c(4nm r 2 ) = c(0) L N,χ ϕ index m ( ) n = r = 0 0 m ( ) m 0 R 0 0

188 178 index 0 ( ) m A k (Γ 0 (N), χ) L N,χ ϕ ( [9] Theorem 3.4. J k,1 (Γ 0 (N) J, χ) A k (Γ (2) 0 (N), χ) L N,χ 1: ϕ N χ J k,1 (Γ 0 (N 1 ) J, χ 1 ) J k,1 (Γ 0 (N) J, χ) ϕ J k,1 (Γ 0 (N 1 ) J, χ 1 ) L N1,χ 1 ϕ L N,χ ϕ ( Saito-Kurokawa lift A k (Γ 0 (N), χ) 2: Gritsenko J k,t (SL 2 (Z) J ) paramodular form of level t Saito-Kurokawa lift ( R. Schmidt Skoruppa-Zagier [21] p, k J k,p (SL 2 (Z) J ) A 2k 2 (Γ 0(p)) (Γ 0(p) Γ 0 (p) normalizer) A k (Γ (2) 0 (N)) Saito Kurokawar lift J k,p (SL 2 (Z)) J k,1 (Γ 0 (p) J ) ([9]) index new form, old form 4. Hecke theory L index L p L ϕ index bad prime ϕ J k,1 (Γ 0 (N) J, χ) p N (6) ϕ k,χ T J (p) = p k 4 g ν (λ,µ) (Z/pZ) 2 χ(a ν )ϕ k,1 [g ν /p] 1 [λ, µ]

189 179 a ν g ν (1, 1) g ν (7) Γ 0 (N)\{g 0,N (p 2 ); gcd(g) = }. gcd(g) g p N U J (p) (6) g ν (8) Γ 0 (N)\Γ 0 (N) ( ) p 2 Γ 0 (N) = b mod p 2 Γ 0 (N) ( ) 1 b 0 p 2. ϕ T J (p) U J (p) λ J (p) F A k (Γ (2) 0 (N), χ) g GSp + (2, Q) M 4 (Z) T = Γ (2) 0 (N)gΓ (2) 0 (N) = ν ( ) Γ (2) Aν B 0 (N) ν. C ν D ν F k,χ T = n(g) 2k 3 ν χ(det(a ν )) det(c ν Z+D ν ) k F ((A ν Z+B ν )(C ν Z+D ν ) 1 ). p N p a + c = b + d p a T S (p a, p b, p c, p d ) = Γ (2) 0 (N) 0 p b p c 0 Γ(2) 0 (N) p d T S (p) = T S (1, 1, p, p), T S (p 2 ) = T S (1, p, p 2, p)+t (1, 1, p 2, p 2 )+ T S (p, p, p, p) p N p U S (p) = Γ (2) 0 (N)diag(1, 1, p, p)γ (2) 0 (N) F T S (p), T S (1, p, p 2, p), U S (p) λ(p), ω(p 2 ), µ(p) F L Spinor L) L(s, F ) = p N (1 µ(p)p s ) 1 p N Q p (p s ) 1 Q p (p s ) = 1 λ(p)p s + (pω(p 2 ) + (p 2 + 1)χ(p) 2 p 2k 5 )p 2s λ(p)χ(p) 2 p 2k 3 3s + χ(p) 4 p 4k 6 4s.

190 180 Theorem 4.1. ϕ J k,1 (Γ 0 (N) J, χ) T J (p) (p N), U J (p) (p N) λ J (p) L N,χ ϕ T S (p), T S (p 2 ), U S (p) L(s, L N,χ ϕ) = L(s k + 1, χ)l(s k + 2, χ) p N (1 λ J (p)p s ) 1 p N (1 λ J (p)p s + χ(p) 2 p 2k 3 2s ) 1 L(s, χ) Dirichlet L ϕ L(s, L N,χ ϕ) = ζ(s)l(s 2k + 3, χ 2 )L(s k + 1, χ)l(s k + 2, χ), ζ(s) p N J k,1 (Γ 0 (N) J, χ) Kohnen plus space A + k 1/2 (Γ 0(4N), χ) A k 1/2 (Γ 0 (4n), χ) Shimura A 2k 2 (Γ 0 (N), χ 2 ) A 2k 2 (Γ 0 (N), χ 2 ) Saito- Kurokawa lift L N,χ (ϕ) T S (p) ϕ Andrianov Eichler-Zagier lift Andrianov [1] [9] References [1] A. N. Andrianov, Modular descent and the Saito-Kurokawa conjecture, Invent Math. 53 (1979), no.3, [2] H. Aoki and T. Ibukiyama, Simple graded rings of Siegel modular forms, differential operators and Borcherds products, Internat. J. Math. 16(2005), [3] T. Arakawa, Saito-Kurokawa lifting for odd weights. Comment. Math. Univ. St. Paul. 49 (2000), no. 2, [4] W. Duke and Ö. Imamoḡlu, A converse theorem and the Saito-Kurokawa lift, Internat. Math. Res. Notices (1996), no.7, [5] M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, 1985, Boston-Basel-Stuttgart. [6] V. Gritsenko, Irrationality of the moduli spaces of polarized abelian surfaces, in Abelian Varieties ed. W. Barth, K. Hulek, H. Lange, de Gruyter(1995), [7] A survey on the new proof of Saito-Kurokawa lifting after Duke and Imamoḡlu, Siegel (1997),

191 181 [8] T. Ibukiyama, On differential operators on automorphic forms and invariant pluriharmonic polynomials, Commentarii Math. Univ. St. Pauli 48(1999), [9] T. Ibukiyama, Saito Kurokawa liftings of level N and practical construction of Jacobi forms, to appear in Kyoto J. Math. Vol. 52 No. 1(2012). [10] T. Ibukiyama, Taylor expansions of Jacobi forms and applications to explicit structures of degree two. to appear in Publication RIMS. [11] J. Igusa, On the graded ring of theta constants, Amer. J. Math. 86(1964), ; (II) ibid. 86(1966), [12] H. Kojima, On construction of Siegel modular forms of degree two, J. Math. Soc. Japan 34(1982), no. 3, [13] J. Kramer, Jacobiformen und Thetareihen, Manusctipta Math. 54 (1986), [14] N. Kurokawa, Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two, Invent. Math. 49(1978), no.2, [15] H. Maaß, Lineare Relationen für die Fourierkoeffizienten einiger Modulformen zweiten Grades, Math. Ann. 232, (1978), [16] H. Maaß, Über eine Spezialshar von Modulformen zweiten Grades, Invent Math. 52(1979) no ; (II) ibid. 53(1979), no ; (III)ibid. 53 (1979), no.3, [17] H. Maaß, Über ein Analogon zure Vermutung von Saito-Kurokawa, Invent. Math. 60 (1980), [18] I. Makino, Dirichlet series corresponding to Siegel modular forms of degree 2, level N, Sci. Papers College Gen. Ed. Univ. Tokyo 28(1978), no.1, Correction: ibid.,29 (1979), no.1, [19] C. Poor ande D. Yuen, A lift into Siegel modular forms over the theta group in degree two and the chiral superstring measure, MPI preprint series 2010(64). [20] I, II, III. [21] N.-P. Skoruppa and D. Zagier, Jacobi forms and a certain space of modular forms. Invent. Math. 94 (1988), no. 1, [22] D. Zagier, Sur la conjecture de Saito-Kurokawa(d après H. Maass), Seminaire Delange Pisot-Poitou, Paris , Progr. Math. 12, Birkhäuser, Boston, Mass. (1981), [23] C. Ziegler, Jacobi forms of higher degree, Abh. Math. Scm. Univ. Hamburg 59 (1989), address: [email protected]

192 [Shn] ( Poisson ) 2 Weil 3 SL(2, R) SO(p, q) 3 1 [Su] [Shn] Heisenberg Weil [Ta] 1.2 z C e[z] = exp(2π 1 1z) z = z 2 π 2 < arg z 1 π 2 2 k Z z k 2 = (z 1 2 ) k t R sgn t = t/ t 1 C 1 = {z C z = 1} R V V S(V ) C (V ) 2 L 2 (V ) m R m Heisenberg W R 2n, W W = X X W ( X, X X, X = X, X = 0, dim R X = dim R X = n, W = X X Department of Mathematics, Rikkyo University, Nishi-Ikebukuro, Tokyo , Japan [email protected] 1

193 183 W ) W w w = x+x (x X, x X ) w = x x X {e 1, e 2,, e n } X {e 1, e 2, e n} e i, e j = δ i,j X x = n i=1 x ie i (x i R) X dx dx = n i=1 dx i X x = n i=1 x i e i (x i R) X d x d x = n i=1 dx i dx i, dx i R Lebesgue f S(X) Fourier ˆf(x ) = f(x)e[ x, x ]dx (x X ) X f S(X ) Fourier ˇf(x) = f(x )e[ x, x ]d x (x X) X Fourier ˇˆf = f f S(X) H(W ) = W R ( (w, t) (w, t ) = w + w, t + t + 1 ) 2 w, w (w, w W, t, t R) Heisenberg H(W ) Z(H(W )) = {(0, t) H(W ) t R} R r R H(W ) L 2 (X) U r [ ( U r (h)f(y) = e r t + 1 )] 2 x, x + y, x f(y + x) (h = (x x, t) H(W ), f L 2 (X)) Heisenberg r R (U r, L 2 (X)) H(W ) 2.2 (Stone-von Neumann ). r R H(W ) (Π, V Π ) Π(0, t) = e[rt] (t R) (Π, V Π ) (U r, L 2 (X)) 2.3 (Schur ). (Π, V Π ), (Π, V Π ) H(W ) (Π, V Π ) (Π, V Π ) H(W )- 1 U 1 U 2

194 R Weil R Weil [Ma] Sp(W ) W Sp(W ) = {σ GL(W ) wσ, w σ = w, w ( w, w W )} GL(W ) W Sp(W ) H(W ) h σ = (wσ, t) (h = (w, t) H(W ), σ Sp(W )) L 2 (X) T Aut(L 2 (X)) f L 2 (X) T T f Hausdorff Mp(W ) U(h σ ) = T 1 U(h) T ( h H(W )) (2.1) (σ, T ) Sp(W ) Aut(L 2 (X)) Mp(W ) compact Hausdorff Stone-von Neumann Mp(W ) (σ, T ) σ Sp(W ) Mp(W ) Sp(W ) 2 σ Sp(W ) wσ = (xa + x c) (xb + x d) ( w = x x W ) [ a End ] R (X), b Hom R (X, X ), c Hom R (X, X), d End R (X ) a b σ = c Hom R (X, X) det c = c d det( e i c, e j ) i,j=1,,n [ ] det a b c 0 σ = Sp(W ) r 0 (σ) Aut(L 2 (X)) c d (r 0 (σ)f)(x) = det c 1 2 X e [ 1 2 xa + x c, xb + x d 1 ] 2 x, x f(xa + x c)d x (f S(X)) r(σ) = (σ, r 0 (σ)) Mp(W ) (1),(2) Ψ: Mp(W ) C 1 1 : (1) Ψ(1, t) = t 2 ( t C 1 ), (2) Ψ(r(σ)) = ( 1) n det c det c ( σ = [ a c b d ] ) Sp(W ) s.t. det c 0. 3

195 185 Sp(W ) = Ker Ψ ϖ : Sp(W ) (σ, T ) σ Sp(W ) Ker ϖ = {(1, ±1)} Sp(W ) Sp(W ) 2 Sp(W ) L 2 (X) ω : Sp(W ) (σ, T ) T Aut(L 2 (X)) (ω, L 2 (X)) Weil [ ] a b Ψ (1),(2) Schur ϖ( σ) = (det c 0) σ c d Sp(W ) ( ɛ ω ( σ) {ɛ ω ( σ)} 2 = ( ) 1) n det c 1 det ω( σ) = c ɛ ω ( σ)r 0 (ϖ( σ)) 2.3 L L X X Z- ψ : Λ C H(W ) Λ = (L L ) R ψ H(W ) (ρ ψ, I Λ (ψ)) I Λ (ψ) θ 1, θ 2 Λ = I Λ (ψ) = {θ C (H(W )) θ(λh) = ψ(λ)θ(h) ( λ Λ)} = Λ\H(W ) 1 vol(x /L ) θ 1 (h)θ 2 (h)dh (2.2) (X/L) (X /L ) θ 1 (x x, 0)θ 2 (x x, 0)dxd x Hilbert H(W ) ρ ψ (h)θ(z) = θ(zh) (h H(W ), θ I Λ (ψ)) vol(x /L ) = d x dh (2.2) 2 X /L Λ\H(W ) H(W ) dh X Z- L L, L = {l X l, l Z ( l L)} M L M, M = {l X l, l Z ( l M )} L M X Z- L M X Z- 4

196 186 Λ 0 = (L L ) R, Λ 1 = (L M ) R H(W ) Λ 1 χ: Λ 1 C χ(λ) = e [ t + 1 ] 2 l, l (λ = (l l, t) Λ 1 ) µ M/L Λ 0 χ µ : Λ 0 C [ χ µ (λ) = e t + 1 ] 2 l, l + µ, l (λ = (l l, t) Λ 0 ) χ µ Λ1 = χ I Λ0 (χ µ ) I Λ1 (χ) H(W )- I Λ1 (χ) = I Λ0 (χ µ ) (2.3) µ M/L θ I Λ1 (χ) θ µ I Λ0 (χ µ ) θ µ 1 (h) = χ µ (λ) 1 θ(λh) #(M/L) λ Λ 1 \Λ 0 θ = µ M/L θµ #(M/L) M/L 2.4. : (1) I Λ1 (χ) (2.3), Λ1 (2) µ M/L f S(X) Θ χµ (f) I Λ0 (χ µ ) Θ χµ (f)(h) = [ e t + 1 ] 2 x, x + µ + l, x f(x + µ + l) (h = (x x, t) H(W )) l L Θ χµ L 2 (X) I Λ0 (χ µ ) H(W )- (3) ν M/L θ I Λ1 (χ) F χν (θ) L 2 (X) [ 1 F χν (θ)(x) = vol(x /M θ((x ν) x, 0)e 1 ] ) X /M 2 x + ν, x d x (x X) F χν (ρ χ, I Λ1 (χ)) (U, L 2 (X)) H(W )- (4) (2),(3) Θ χµ F χν F χν Θ χν = id L 2 (X), F χν Θ χµ = 0 (ν µ) Fourier (2.3), Λ1 I Λ0 (χ µ ), Λ0 Schur (U, L 2 (X)) (ρ χ, I Λ1 (χ)) H(W )- Θ χµ 5

197 Sp(L M ) (L M )γ = L M, χ(λ γ ) = χ(λ) ( λ Λ 1 ) γ Sp(W ) Sp(W ) Sp(L M ) = ϖ 1 (Sp(L M )) µ M/L f S(X) Sp(W ) ϑ f ( σ; µ) ϑ f ( σ; µ) = Θ χµ (ω( σ)f)(0, 0) = l L(ω( σ)f)(x + µ + l) 2.5 ( ). γ Sp(L M ) µ M/L ϑ f ( γ σ; µ) = C γ (µ, ν)ϑ f ( σ; ν) ( f S(X)) (2.4) ν M/L C γ (µ, ν) (ν M/L) M/L = {µ 1, µ 2,, µ N } (N = #(M/L)) C γ = (C γ (µ i, µ j )) i,j=1,,n [ C γ1 γ 2 = ] C γ1 C γ2 ( γ 1, γ 2 Sp(L M )) ϖ( γ) = a c b d det c 0 C γ (µ, ν) C γ (µ, ν) = ɛ ω( γ) det c 1 2 vol(x /M ) l L/M c e [ 1 2 µ + l, (µ + l)ac 1 µ + l, νc ] 2 ν, νc 1 d (2.5) c x 1 c, x 2 = x 2 c, x 1 ( x 1, x 2 X ) Hom R (X, X). Sp(L M ) γ Ξ γ Aut(I Λ1 (χ)) (Ξ γ θ)(h) = θ(h γ ) (θ I Λ1 (χ)) Ξ γ ρ χ (h γ ) = ρ χ (h) Ξ γ ( h H(W )) γ Sp(L M ) γ = ϖ( γ) Ξ γ 1 Θ χµ ω( γ) (U, L 2 (X)) (ρ χ, I Λ1 (χ)) H(W )- C γ (µ, ν) (ν M/L) Ξ γ 1 Θ χµ ω( γ) = C γ (µ, ν)θ χν (2.6) ν M/L ω( σ)f ( σ Sp(W ), f S(X)) (0, 0) H(W ) (2.4) γ 1, γ 2 Sp(L M ) γ 1 = ϖ( γ 1 ), γ 2 = ϖ( γ 2 ) Ξ (γ1 γ 2 ) 1 Θ χ µi ω( γ 1 γ 2 ) = Ξ γ 1 2 (Ξ γ 1 1 Θ χµi ω( γ 1 )) ω( γ 2 ) (i = 1, 2,, N) 6

198 188 (2.6) N j=1 C γ 1 γ 2 (µ i, µ j )Θ χµj Ξ γ 1 2 ( N ) C γ1 (µ i, µ k )Θ χµk ω( γ 2 ) = k=1 C γ1 γ 2 (µ i, µ j ) = N = N N C γ1 (µ i, µ k ) C γ2 (µ k, µ j )Θ χµj k=1 j=1 ( N N ) C γ1 (µ i, µ k )C γ2 (µ k, µ j ) j=1 k=1 Θ χµj k=1 C γ 1 (µ i, µ k )C γ2 (µ k, µ j ) C γ1 γ 2 = C γ1 C γ2 L 2-1 f S(X) V f = N CΘ χµi (f) I Λ1 (χ) i=1 C γ V f 2 {Ξ γ 1Θ χµi (ω( γ)f)} N i=1 {Θ χ µi (f)} N i=1 (2.5) (2.6) ω( γ 1 )f (f S(X)) Ξ γ 1Θ χµ (f) = C γ (µ, ν)θ χν (ω( γ 1 )f) 2.4(4) ν M/L F χν (Ξ γ 1Θ χµ (f)) = C γ (µ, ν)ω( γ 1 )f. (2.7) (2.5) 3 SL(2, R) SO(Q) 3.1 (SL(2, R), O(Q)) ( ) 1 J = SL(2, R) R 2, J r, r J = 1 rj t r (r, r R 2 ) σ SL(2, R) rσ, r σ J = r, r J (σ SL(2, R)) Q M n (Q) (p, q) g Q GL(n, Q) ( ) 1 p Q = g Q 1 q t g Q p > 0 R n (, ) Q (x, x ) Q = xq t x (x, x R n ) (, ) Q O(Q) O(Q) = {g GL(n, R) (xg, x g) Q = (x, x ) Q ( x, x R n )} 7

199 189 = {g GL(n, R) gq t g = Q} W,,, X, X, e i, e i W = R n R R 2 W, x r, x r = (x, x ) Q r, r J (x r, x r W ) W W = X X X = R n R (1, 0), X = R n R (0, 1). R X x (1, 0) x R n, X x (0, 1) x R n X, X R n X {e 1, e 2,, e n } R n ( e i i 1 0 R n ) X {e 1, e 2,, e n} e i = e iq 1 dx R n Lebesgue d x = det Q dx L 2 (X) Weil Q ω( σ) r 0 (σ) ω( σ, Q) r 0 (σ, Q) SL(2, R) O(Q) (x r)σ = x (rσ) (x r W, σ SL(2, R)), (x r)g = (xg) r (x r W, g O(Q)) W Sp(W ) (SL(2, R), O(Q)) Sp(W ) SL(2, R) = {σ Sp(W ) σg = gσ ( g O(Q))}, O(Q) = {g Sp(W ) σg = gσ ( σ SL(2, R))} SO(Q) = SL(n, R) O(Q) ϖ 1 (SL(2, R)) ϖ 1 (SO(Q)) ϑ f ( σ; µ) SL(2, R)SO(Q)( Sp(W )) 3.1. ϑ f ( σ; µ) SL(2, R)SO(Q) SL(2, R)O(Q) 8

200 X, X R n End R (X), ( Hom R (X, ) X ), Hom R (X, X), End R (X ) a b M n (R) σ = SL(2, R) g O(Q) Sp(W ) c d [ ] [ ] a1 n b1 n g O n, c1 n d1 n O n g c ( M n (R) ) det c = (det Q) 1 det c a b c 0 σ = SL(2, R) f S(X) c d [ ] (r 0 (σ, Q)f)(x) = c n a(x, x)q 2(x, y) Q + d(y, y) Q 2 det Q e f(y)dy R n 2c ( ) a b σ = SL(2, R) g O(Q) 0 d r 0 (σ, Q) r 0 (g, Q) r 0 (σ, Q) = r 0 (σj, Q) r 0 ( J 1, Q ), r 0 (g, Q) = r 0 (gj, Q) r 0 ( J 1, Q ) Fourier f S(X) [ ] (r 0 (σ, Q)f)(x) = a n ab 2 e 2 (x, x) Q f(xa), (r 0 (g, Q)f)(x) = f(xg) GL(n, R) L 2 (X) R (R(g)f)(x) = det g f(xg) (g GL(n, R), f L 2 (X)) r 0 (g, Q) = R(g) ( g O(Q)) σ SL(2, R) r 0 (σ, Q) r 0 (σ, gq t g)r(g) = R(g)r 0 (σ, Q) (g GL(n, R)) (3.1) 3.3 ϖ 1 (SL(2, R)) ϖ 1 (SO(Q)) ϖ 1 (SL(2, R)) H = {z = u + ( ) a b 1v u, v R, v > 0} σ = SL(2, R) z H c d 9

201 191 j(σ, z) = cz + d, σ z = az + b cz + d, ɛ(σ) = ( 1) 1 2 (c > 0 ), ( 1) 1 sgn(d) 2 (c = 0 ), ( 1) 1 2 (c < 0 ). ( ) 1 p Q = g t Q g Q g Q GL(n, R) 1 q R R = g t Q g Q z = u + 1v H Q z = uq + 1vR 0 k P k (x) X k : 1 k = 0, rq t x (r C n s.t. r(q R) = 0) k = 1, (3.2) r c r(rq t x) k (c r C, r C n s.t. r(q R) = 0, rq t r = 0) k 2. ( p = 1 k 1 ) [ ] F z (x) = e 2 xq z t x P k (x) S(X) r 0 (σ, Q)F z (x) = ɛ(σ) p q j(σ, z) q p 2 k j(σ, z) q F σ z (x) σ SL(2, R). g GL(n, R) [ R(g)F z (x) (3.2) ] k 1 P k (x) R(g)F z(x) = e 2 x(gq t g) t z x det g P k (x) (3.1) ( ) 1 p Q = R = 1 n 1 ( ) q a b σ = c = r 0 (σ, Q) c d c r 0 (σ, Q) [ ] c n a(x, x)q 2(x, y) Q + d(y, y) Q 2 F z (y)dy 2c R n e = c n 2 (v 1u 1d/c) p 2 (v + 1u + 1d/c) q 2 j(σ, z) k F σ z (x) (3.3) k = 0 (3.3) Cauchy (3.3) e πt2 dt = 1 k > 0 (3.3) k = 0 (3.3) ( ) 1 (2π 1) k (az + b) P k k,,, (x = (x 1, x 2,, x n ) ) x 1 x 2 x n ( ) 10

202 192 ( c 0 σ = a c ) b d SL(2, R) {ɛ(σ) p q } 2 = ( 1) n det (c1 n ) det (c1 n ) Ψ (1),(2) ( σ, ɛ(σ) q p r 0 (σ, Q)) Sp(W ) = Ker Ψ Schur σ, τ SL(2, R) (ɛ(σ) q p r 0 (σ, Q)) (ɛ(τ) q p r 0 (τ, Q)) = c(σ, τ)ɛ(στ) q p r 0 (στ, Q) c(σ, τ) c(σ, τ) = { j(στ, 1) 1 2 j(σ, τ 1 ) 1 2 j(τ, 1) SL(2, R) {±1} } p q (σ, t) (σ, t ) = ( σσ, tt c(σ, σ ) ) ((σ, t), (σ, t ) SL(2, R) {±1}) (n ) ι 1 : SL(2, R) {±1} (σ, t) (σ, tɛ(σ) q p r 0 (σ, Q)) SL(2, R) ϖ 1 (SO(Q)) Ψ (2) g O(Q) Ψ(g, r 0 (g, Q)) = Ψ(gJ, r 0 (gj, Q))Ψ(J 1 (, r 0 J 1, Q ) ) { = ( } { 1) n det ( g) det ( } 1) n det 1 n ( g) det 1 n = det g 3.4. g SO(Q) (g, r 0 (g, Q)) Sp(W ) = Ker Ψ ι 2 : SO(Q) g (g, r 0 (g, Q)) Sp(W ) ϖ : Sp(W ) Sp(W ) SO(Q) 3.5. ϖ SO(Q) O(Q) 11

203 L R n Z- L = {l R n (l, l ) Q Z ( l L)} L L X, X R n L L M L L M Z- f S(X) µ L /L ϑ f ( σ, g; µ) = l L ω(ι 1 ( σ)ι 2 (g))f(µ + l) ( σ SL(2, R), g SO(Q)) 2.5 : 3.6. (1) γ 2 SO(Q) GL(n, Z) ϑ f ( σ, γ 2 g; µ) = ϑ f ( σ, g; µγ 2 ). ( ) a b (2) γ 1 = SL(2, Z) ab(l, l) c d Q cd(l, l) Q 0 mod 2 γ 1 = (γ 1, ε) SL(2, R) ϑ f ( γ 1 σ, g; µ) = C γ1 (µ, ν)ϑ f ( σ, g; ν) ν L /L ( l L) f S(X) C γ1 (µ, ν) : C γ1 (µ, ν) ε( sgn c (q p) 1) 2 [ ] a(µ + l, µ + l)q 2(µ + l, ν) Q + d(ν, ν) Q e c n 2 det Q vol(r n /L) 2c l L/cL = εδ µ,aν ( [ ] 1 sgn d (q p) ab 1) 2 e 2 (µ, µ) Q { 1 (µ = µ ), δ µ,µ = 0 (µ µ ) (c 0 ), (c = 0 ). C γ1 (µ, ν) : 3.7. L Z- {l 1, l 2,, l n } D = det((l i, l j ) Q ) 3.6(ii) c 2Z, cl L, cd 0, c(l, l) Q 0 mod 2 ( l L ) ε( 1 sgn d (p q) sgn 1) 2 c C γ1 (µ, ν) [ ] ab δ µ,dν e 2 (µ, ν) Q ( 2c ε n d ( 1 sgn c) n d ( 2c ) n ( ) D d (d < 0 ), = [ ] ) ab n ( ) D δ µ,dν e 2 (µ, ν) Q ε n d (d > 0 ) d d { 1 (d 1 mod 4 ), ( ) ε d = 1 (d 3 mod 4 ) ([Shm] ) 12

204 H SO(Q) 3.4 [Shn] SL(2, R) SO(2) SO(2) = {(κ t, ε) t R, ε {±1}} ( ) cos t sin t κ t = sin t cos t (( v ) ) u/ v σ z = 1/, 1 v m f S(X) SO(2) z = u + 1v H ω(ι 1 (κ t, ε))f = ε(e 1t ) m 2 f ( (κt, ε) SO(2)) (3.4) µ L /L ( m = p q + 2k f = F z (3.4) ) H SO(Q) (z, g; µ) θ H f θ H f (z, g; µ) = v m 2 θf ( σ z, t g 1 ; µ) 3.8. γ 2 SO(Q) GL(n, Z) 3.6 (2) γ 1 SL(2, Z) j(γ 1, z) m 2 θ H f (γ 1 z, γ 2 g; µ) = ν L /L C (γ1,1)(µ, ν)θ H f (z, g; νt γ 1 2 ) [Ma]. Weil Howe duality. ( ) [Shm] Goro Shimura. On modular forms of half integral weight. Ann. of Math. 97, pp , [Shn] Takuro Shintani. On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J., Vol. 58, pp , [Su]. Oda lift. ( ) [Ta]. Weil theta. 4, pp ,

205 195 Oda Lift [5] weight O(2,n 2) lifting Weil Shintani [6], Niwa [4] O(2, 1) 0 SL 2 (R) O(2, 2) 0 SL 2 (R) SL 2 (R) n =4 Doi-Naganuma lift [1], [3] weight 2k 2 weight k 2 Siegel lifting (Saito- Kurokawa lift) Zagier [9] Jacobi [7] Jacobi Oda lift 1, 2 3 Oda lift Oda [5] Jacobi 4 Fourier Maass lift Oda lift Hecke Jacobi Q 2 [5] Q 1 Q n x, y M n,1 Q(x, y) := t xqy, Q[x] :=Q(x, x) z C e[z] =e 2πiz P { 1 P δ(p ):= 0 P δ(a = b) Kronecker delta δ a,b S m even integral i.e. 1 1 Q 0 := S, Q 1 := Q 0, Q = Q 2 := Q Q (2,m+2) V 0 := R m, V 1 := R V 0 R = R m+2, V = V 2 := 1 R V 1 R = R m+4

206 196 L 0, L 1, L = L 2 L i Q i L i = Q 1 i L i (i =0, 1, 2) { } G i = O(Q i ) 0 0 := g i GL m+2i (R) t g i Q i g i = Q i (i =0, 1, 2) G i G = G 2 G 0 G 1 G V 1,C = V 1 R C = C m+2 D := { Z = τ } w V 1,C Q1 [Im Z] > 0, Im τ>0 Z 0 = i 0 z i Q 1 [Im Z] > 0 Z V 1,C D D disjoint union g G, Z D Z D Z := Q 1[Z]/2 Z V C, g Z := (g Z ) J G (g, Z) 1 G D Z g Z G D J G (g, Z) Z 0 D G K Z D G/K = D, K = SO(2) SO(m +2) dz := (Q 1 [Im Z]/2) (m+2) d(rez) d(imz), d(rez), d(imz) :Lebesgue D G x V 1,y V 0 1 t xq 1 Q 1 [x]/2 n(x) := 1 m+2 x 1 G G, n 1 (y) := 1 t yq 0 Q 0 [y]/2 1 m y 1 G 1 g = n(x)n 1 (y)diag(a, b, 1 m,b 1,a 1 ) k (x V 1, y V 0, a,b > 0, k K) dg := a (m+3) b (m+1) dx dy da db dk, vol(k) =1 G Haar D ϕ ϕ(g Z 0 ) dg = 1 ϕ(z) dz G 2 D 2

207 G Γ, Γ Γ := G GL m+4 (Z) Γ := {γ Γ (γ 1)L L} Γ Γ k N D F { (i) F (γ Z ) =J G (γ,z) k F (Z) for γ Γ (ii) Sup g G F gr (g) < F gr (g) :=F (g Z 0 ) J G (g, Z 0 ) k Γ weight k S k (Γ ) Γ S k (Γ ) 1.1 F S k (Γ ) F (Z) = a F (η) e[q 1 (η, Z)] η L 1 iη D Fourier τ Fourier F ( τ w ) = F n (z,w) e[nτ], F n (z,w) = a F ( a α ) e[az S(α, w)] z n=1 n a Z, α L 0 F 1, F 2 S k (Γ ) Petersson F 1,F 2 k := F 1 (Z) F 2 (Z)(Q 1 [ImZ]/2) k dz Γ \D 2. Jacobi 2.1. S m { } even integral H S := [ξ,η,ζ] ξ,η V 0, ζ R [ξ,η,ζ] [ξ,η,ζ ]:=[ξ + ξ,η+ η,ζ + ζ + S(ξ,η )] [0, 0, 0] [ξ,η,ζ] 1 =[ ξ, η, ζ + S(ξ,η)] H S Z S := {[0, 0,ζ] ζ R} G := SL 2 (R) H S g 1 [ξ,η,ζ]g := [ξ,η,ζ ], (ξ,η ):=(ξ,η)g, ζ := ζ S(ξ,η)/2+S(ξ,η )/2 H S G G S := H S G Jacobi Z S H S 3

208 198 ( ) a b Jacobi G S G g = G := SL c d 2 (R) ( ) ι(g) := g 1 n 2, g := J 1 t g 1 a b J = g c d ι(g) G ξ,η V 0, ζ R 1 0 t ηs S(ξ,η) ζ S[η]/2 0 1 t ξs S[ξ]/2 ζ ι([ξ,η,ζ]) := 1 m ξ η G ι([ξ,η,ζ]g) :=ι([ξ,η,ζ])ι(g) ι : G S G ι G S G Jacobi G S D S := H C m g (z,w) := (g z,wj(g, z) 1 + ξg z + η) (g[ξ,η,ζ]g) ( ) a b g = G g z := (az+b)(cz+d) 1,j(g, z) :=cz+d c d Z 0 =(i, 0) Z S SO(2) k, n N G S D S J k,n ([ξ,η,ζ]g, (z,w)) := j(g, z) k e [ { n ζ + ( ) c S[w] S(ξ,w) 2 J k,n G S D S 2.2. Jacobi j(g, z) 1 g z }] S[ξ] 2 J k,n (g g,z)=j k,n (g,g Z ) J k,n (g,z) (g,g G S, Z D S ) Γ := SL 2 (Z) G Γ S := {[ξ,η,ζ] ξ,η L 0, ζ Z} Γ D S f 2 { (i) f(γ Z ) =Jk,n (γ,z) f(z) γ Γ S, Z D S (ii) Sup g GS f(g Z 0 )J k,n (g,z 0 ) 1 < Γ S weight k, index n n S Jacobi S k,n (Γ S ) D S = H C m dz := dx dy y 2 dξ dη, Z =(z,w) =(x + iy, ξz + η) H 4

209 199 dξ, dη V = R m Lebesgue dz G S = H S G Jacobi S k,n (Γ S ) Petersson f 1,f 2 k,n := f 1 (Z) f 2 (Z) y k e 2πnyS[ξ] dz Γ S \D S Hilbert { ( ) 1 x f S k,n (Γ S ) Γ S, := [0,η,ζ] η L0, ζ,x Z} G 0 1 S 2.1 f S k,n (Γ S ) f(z) = a f (a, α) e a,α (Z), e a,α (Z) :=e[az + S(α, w)] a N,α L 0 an S[α]/2>0 Fourier 2.1 g =[ξ,η,ζ]g G S, Z = τ w z D g Z := τ w z (z,w )=g (z,w), J(g, Z) =j(g, z) τ = τ + ζ ( c 2 S[w] S(ξ,w))j(g, z) 1 + g z 2 S[ξ] Γ S Γ F S k (Γ ) n-th Fourier-Jacobi F n S k,n (Γ S ) 2.3. z H C m h h(w + ξz + η) =e[ z 2 S[ξ] S(ξ,w)] h(w) ξ,η L 0 Θ S,z α L 0 θ α (z,w) := e[ z S[α + l]+s(α + l, w)] 2 l L 0 H Θ S,z 2.2 θ α α L 0 /L 0 {θ α (z,w) α L 0 /L 0} Θ S,z θ α Γ = SL 2 (Z) Shintani [6] Proposition 1.6, [2] ( ) a b 2.3 γ = Γ c d θ α (γ (z,w) ) =ε(γ) m j(γ,z) m/2 J 0,1 (γ,(z,w)) c α,β (γ)θ β (z,w) 5 β L /L

210 200 δ α,aβ e[abs[α]/2] (c =0) [ ] c α,β (γ) := as[α + l] 2S(α + l, β)+ds[β] (det S) 1/2 c m/2 e (c 0) 2c l L 0 /L 0 c { e[sgn(c)/8] (c 0) ε(γ) := e[(1 d)/8] (c =0) U(γ) :=ε(γ) m (c α,β (γ)) det S = L /L 2.2 N S N S 1 even integral Γ 1 (2N) U(γ) f S k,1 (Γ S ) z H f(z, ) Θ S,z ( ) f(z,w) = ϕ α (z) θ α (z,w) α L 0 /L 0 f (ϕ α ) α L 0 /L 0 S k,1 (Γ S ) = S k m/2 (Γ, U) (ϕ α (γ z ) α L 0 /L 0 = j(γ,z) k m/2 U(γ)(ϕ α (z)) α L 0 /L 0 γ Γ 2.4. Poincaré (a, α) Z L 0 Δ a,α := a S[α]/2 > 0 f a,α (Z) := J k,1 (γ,z) 1 e a,α (γ Z ) (Z D S ) γ Γ S, \Γ S Poincaré 2.4 k>m+2 f a,α (Z) D S f a,α S k,1 (Γ S ) (1) f S k,1 (Γ S ) f,f a,α k,1 = A S,k Δ (k 1 m/2) a,α a f (a, α), A S,k =(dets) 1/2 2 (k 1) (2π) (k 1 m/2) Γ(k 1 m/2). S k,1 (Γ S ) Poincaré f a,α (2) (a, α), (b, β) Z L 0, Δ a,α, Δ b,β > 0 Δ (k 1 m/2) a,α a fb,β (a, α) =Δ (k 1 m/2) b,β a fa,α (b, β). 2.3 (a, α), (b, β) Z L 0 (a, α) (b, β) α β (mod L 0 ) Δ a,α = Δ b,β 6

211 201 f S k,1 (Γ S ) Fourier Poincaré Poincaré Fourier 2.5 (a, α) Z L 0,Δ a,α > 0 f a,α (b, β) (Δ b,β > 0) Fourier a fa,α (b, β) =C + ((a, α); (b, β)) + ( 1) k C + ((a, α); (b, β)) C + ((a, α); (b, β)) = δ((a, α) (b, β)) + 2π( i) k (det S) 1/2( Δ b,β Δ a,α ) (k 1 m/2)/2 H c ((a, α); (b, β)) = H c ((a, α); (b, β))c 1 m/2 J k 1 m/2 ( 4π Δa,α Δ b,β ) c c=1 ξ L 0 /L 0 c d Z/cZ (c,d)=1 [ 1 {a 0 e c 2 S[ξ]+a 0S(ξ,α)+a 0 a + db S(ξ + α, β) }] (c, d) =1 a 0 d b 0 d =1 a 0,b 0 Z J Bessel J ν (z) :=(z/2) ν n=0 ( 1) n (z/2) 2n n!γ(ν + n +1) 3. Oda lift 3.1. z = x + iy H, k N V = R m+4 f z,k 1 2 f z,k (v) :=Q(Z0,v) k e[q z [v]/2], Q z := xq + iyr, R := S 1 2 θ k (z,g; μ) := l L y (m+2)/2 f z,k (g 1 (l + μ)) ( ) μ L a b γ = Γ c d θ k (γ z,g; μ) =ε(γ) m j(γ,z) k m/2 c μ,ν(γ) ν L /L c μ,ν(γ)θ k (z,g; ν) δ(μ νa L) e[abq[μ]/2] (c =0) [ aq[μ + r] 2Q(μ + r, ν)+dq[ν] ] det Q 1/2 c (m+4)/2 e (c 0) 2c r L/Lc 7

212 202 [6] Proposition 1.6 π : L L 0 Z =(z,w) D S, g G Θ k (Z, g) := θ k (z,g; μ) θ π(μ) (Z) μ L /L = y (m+2)/2 μ L f z,k (g 1 μ) θ π(μ) (Z) 3.1 (1) Θ k (Z, γgκ) =J G (κ, Z 0 ) k Θ k (Z, g) for γ Γ, κ K. (2) Θ k (γ Z,g)=J k,1 (γ,z) Θ k (Z, g) for γ Γ S Oda lift k >2m +4 f S k,1 (Γ S ), F S k (Γ ) (ιf)(g) := f(z) Θ k (Z, g) y k e 2πyS[ξ] dz Γ S \D S (ρf )(Z) := F (g) Θ k (Z, g) dg Γ \G F (g) :=F (g Z 0 ) J G (g, Z 0 ) k F G ι(f), ρ(f ) Γ,Γ S Petersson adjoint ι(f),f k = f,ρ(f ) k,1 (f S k,1 (Γ S ), F S k (Γ )) Fourier k ι(f) S k (Γ ), ρ(f ) S k,1 (Γ S ) 3.3. ρf Fourier F S k (Γ ) ρf (ρf )(γ Z ) =J k,1 (γ,z)(ρf )(Z) (γ Γ S ) v V G v := {h G hv = v}, K v := K G v, Γv := Γ G v G v Haar d v h G v \G d vġ Φ(g) dg = d vġ Φ(hg) d v h G G v\g G v (ρf )(Z) = θ π(μ) (Z) {μ} Γ Γ μ \G y (m+2)/2 f z,k (g 1 μ) F (g) dg {μ} Γ μ L Γ Q[μ] 0 = Φ(hg) d μ h =0 Γμ \Gμ 8

213 203 μ L i.e. n 2 μ L n γ Γ γμ = η a,α := ρf (Z) = 0 η a,α, η a,α = a α (a, α) Z L a Z, α L 0 /L 0 n N Δ a,α>0 θ nα (Z) n k (m+2) A((a, α); n 2 z) A((a, α); z) = y (m+2)/2 f z,k (g 1 ηa,α) F (g) dg Γη a,α \G G a,α := G η a,α (1,m+2) O( 1 T 1 ( ) ) 0 S Sα, T = t αs 2a h = 1 t yt T[y]/2 1 m+1 y 1 t 1 m+1 k (k K a,α ) t 1 G a,α Haar d a,α h := t (m+2) dt dx dk ( K a,α dk =1 ) 3.2 k>2m +4, F S k (Γ ) (1) ρf S k,1 (Γ S ). (2) (a 0,α 0 ) Z L 0, Δ a 0,α 0 > 0 a ρf (a 0,α 0 )=c(ρ) a,n N, α L 0 /L 0 Δ a0,α 0 =n 2 Δ a,α, α 0 nα L 0 n k m 2 Δ (k m 1)/2 a,α I a,α I a,α := F (hg a,α ) d a,α h, c(ρ) :=i k (det S) 1/2 2 k 1 m/2. Γa,α \Ga,α 3.4. ιf Fourier (a, α) Z L 0, Δ a,α > 0 Ω a,α (Z) := l L π(l)+α L 0 Q[l]/2=Δ a,α Q(Z,l) k = n Z Ω (n) a,α (Z) 9

214 204 Ω a,α (n) (Z) := Ω (0)+ a,α (Z) := b Z, β L 1 π(β)+α L 0 bn+q 1 [β]/2=δ a,α b Z, β L 1 π(β)+α L 0 Q 1 [β]/2=δ a,α, iβ D ( n ) k 2 Q 1[Z]+Q 1 (Z,β)+b ( ) k Q 1 (Z,β)+b Ω a,α(z) + :=Ω a,α (0)+ (Z)+ n 1 Ω(n) a,α(z) Ω a,α (Z) =Ω + a,α(z)+( 1) k Ω + a, α (Z) 3.3 (a, α) Z L 0, Δ a,α > 0 (1) Poincare f a,α ι Ω a,α (ιf a,α ) dm (Z) :=(ιf a,α )(g) J G (g, Z 0 ) k = c(ι) Ω a,α (Z) (Z = g Z 0 ). c(ι) :=2 m/2 (det S) 1/2 π k Γ(k). (2) Ω a,α S k (Γ ) Ω + a,α ν L 1 (iν D) Fourier C+ a,α(ν) Bessel C + a,α(ν) =C (0)+ a,α (ν)+ C a,α (0)+ (ν) := ( 2πi)k Γ(k) (2π) k+1 n 1+m/2 Γ(k)(det S) 1/2 n=1 Δ a,α Q 1 [ν]/2) J k 1 m/2 ( 4π n r N,ν L 1 r Q 1 [νr 1 ]/2=Δ a,α π(νr 1 )+α L 0 r k 1 ( Q1 [ν] 2Δ a,α ) (k 1 m/2)/2 λ L 1 /L 1n Q 1 [λ]/2 Δ a,α nz π(λ)+α L 0 e[ Q 1 (ν, λ)/n] 3.5. Zagier identity Fourier Maass Zagier [8] Theorem 3, Oda [5] Theorem Z D S, Z D Δa,α k 1 m/2 Ω a,α (Z) e a,α (Z) = (a,α) Z L 0 Δ a,α>0 Δ k 1 m/2 b,β Ω (0)+ b,β (b,β):/ (Z) f b,β(z) (b, β) 2.3 f b,β Fourier e a,α (Z) e[q 1 (ν, Z)] J k 1 m/2 ( )

215 Maass type lift 4.1. Maass N N, f S k,1 (Γ S ) D S V N f (V N f)(z,w) :=N k 1 j(b,z) k e [ cn2 ] S[w]j(B,z) 1 f(b z,wnj(b,z) 1 ) T (N) = B Γ \T (N) { ( ) a b B = M c d 2 (Z) } det B = N B z := az + b, j(b,z) =cz + d for B = cz + d V N f S k,n (Γ S ) ( ) a b T (N) c d 4.1 f S k,1 (Γ S ) D I(f) I(f)( τ w ) = (V N f)(z,w) e[nτ] z N=1 { } = r k 1 a f (abr 2,αr 1 ) e[az + S(α, w)+bτ] a,b N,α L 0 r N a,b rz α L 0 r (1) I(f) S k (Γ ) I Maass type lift (2) I S k,1 (Γ S ) S k (Γ ) I Maass space { F S k (Γ ) a F ( a α ) = b r r N a,b rz, α L 0 r Γ Γ S, Γ1 J M := 1 m J k 1 a F ( abr 2 αr 1 1, J := ) for a α b ( ) L 1 V N Γ S Fourier Γ1 M τ z 4.2. Oda } 4.2 k>2m +4 f S k,1 (Γ S ) (ιf) dm (Z) :=(ιf)(g) J G (g, Z) k =2c(ρ) I(f)(Z) (Z = g Z 0 ). 11

216 206 Zagier Identity 3.4 e a,α (Z) (a, α) Z L 0, Δ a,α > Ω a,α (Z) = Δ k 1 m/2 a,α Ω a,α (Z) = (b,β):/ = (b,β):/ = ( 2πi)k Γ(k) (b,β)/ Ω (0)+ b,β (Z) a f a,α (b, β) ( 2πi) k Γ(k) ν L 1,iν D t ν=(a, t α,b) Ω (0)+ b,β (Z) Δk 1 m/2 ν L 1,iν D r N, ν L 1 r Q 1 [νr 1 ]=b S[β]/2 π(νr 1 )+β L 0 r Nνr 1 L 1 b,β a fb,β (a, α). r k 1 e[q 1 (ν, Z)] a fa,α (b, β) r k 1 a fa,α (abr 2, αr 1 ) e[q 1 (ν, Z)] 5. Hecke m even integral S maximal g M m (Z) GL m (Q) S[g 1 ]:= t g 1 Sg 1 even integral g GL m (Z) Jacobi Q G Q { } G Q := g GL m+4 (Q) t gqg = Q v Q v G v adele G A R G Jacobi Hecke L p G S,p = G S (Q p ) compact K S,p { } K S,p := [ξ,η,ζ]g ξ,η L 0,p := L 0 Z Z p, ζ Z p, g SL 2 (Z p ) G S,p Z S,p := {[0, 0,ζ] ζ Q p } Q A /Q basic character χ = v χ v χ (x) =e[x] G S,p K S,p C φ φ([0, 0,ζ]g) =χ p (ζ)φ(g) for ζ Q p, g G S,p Z S,p \supp φ compact H S,p convolution (φ 1 φ 2 )(g) := φ(g g 1 1 ) φ 2 (g 1 ) dg 1 Z S,p \G S,p vol(z S,p \Z S,p K S,p )=1 12

217 207 C-algebra φ 0,p G S.p 1 support Z S,p K S,p S Q p Witt ν p m =2ν p +n 0,p n 0,p 0 n 0,p 4 S maximal L 0,p := {x L 0.p S[x]/2 Z p} L 0,p Z p lattice L 0,p /L 0,p F p = Z p /pz p p 0 p 2 φ p,φ 0,p H S,p ( ) ( p p supp φ p = Z S,p K S,p p 1 K S,p, φ p ( p 1 ) )=1 supp φ 0,p = Z S,p K S,p {[0,η,0] η L 0,p}K S,p, φ 0,p([0,η,0]) = p p (η L 0,p) 2 H S,p H S,p H S,p p =0, 1 H S,p φ 0,p φ 0,p = { φ 0,p p =0, 1 (1 p 1 )φ 0,p + p 1 φ 0,p p =2 C-algebra λ p : H S,p C L L p(λ p ; s) L p (λ p ; s) := {1 (λ p (φ p )p (1+m/2) p p n0,p/2 + p 1+n0,p/2 )p s + λ p (φ 0,p) 1 p 2s} 1 { } (1 χ S (p)p s ) 1 m :even B S,p (p s ), 1 m :odd 1 p =0or(n 0,p, p )=(2, 1) 1+p 1/2 T (n 0,p, p )=(1, 1) (1 + pt )(1 + T ) (n 0,p, p )=(2, 2) B S,p (T )= 1 p 1/2 T (n 0,p, p )=(3, 1) (1 + p 1/2 T )(1 p 1/2 T ) (n 0,p, p )=(3, 2) (1 pt )(1 T ) (n 0,p, p )=(4, 2) m χ S Q p ( ( 1) m/2 det S)/Q p p< H S,p convolution Jacobi S k,1(γ S ) S k,1 (Γ S ) f φ = λ f (φ)f for φ p< H S,p L(f; s) := p L p(λ p ; s) f L 5.1 (1) f S k,1 (Γ S ) Hecke (a, α) Z L ( ) 0 S Sα T := t T maximal even integral αs 2a a f (an 2,αn)n (s+k 1 m/2) n=1 13

218 208 = a f (a, α) L(f; s) { ζ(2s) 1 L(χ T ; s +1/2) 1 m :evem m :odd } B T,p (p (s+1/2) ) 1. p< (2) L(f; s) { 2 s π 3s/2 (det S) s/2 Γ(s + k 1 m/2)γ((s + a)/2) m :even L (f; s) := (2π) s (2 1 det S) s/2 Γ(s + k 1 m/2) m :odd a m 0, 2(mod4) 1, 0 ξ(f; s) :=L (f; s) L(f; s) s { } 1 m 1, 3 (mod 8) ξ(f; s) = ξ(f;1 s) 1 otherwise 5.2. Hecke L (2,m+2) Q G = O(Q) Q p G p compact K p K p K p := G p GL m+4 (Z p ) K p := {u K p (u 1)L p L p} G p K p compact support H(G p,k p ) convolution C-algebra H(G p,k p ) = C[X ± 1,...,X± ν p+2 ]W νp+2 W νp+2 X 1,...,X νp+2 X i X 1 i Weyl Λ p H(G p,k p ) L L p (Λ p ; s) ν p+2 L p (Λ p ; s) Λ p ( (1 X i p s )(1 Xi 1 p s )) = i=1 1 (n 0,p, p )=(0, 0) or (1, 0) 1+p s+1/2 (n 0,p, p )=(1, 1) (1 p 2s ) 1 (n 0,p, p )=(2, 0) (1 p s ) 1 (n 0,p, p )=(2, 1) (1 p s ) 1 (1 + p 1 s ) (n 0,p, p )=(2, 2) (1 p s 1/2 ) 1 (n 0,p, p )=(3, 1) (1 p s 1/2 ) 1 (1 + p s+1/2 ) (n 0,p, p )=(3, 2) (1 p s ) 1 (1 p s 1 ) 1 (n 0,p, p )=(4, 2) S maximal G 1,A = G 1,Q G 0 1, 14 p< K 1,p

219 209 F S k (Γ ) G A p H(G p,k p ) convolution S k (Γ ) F Φ=Λ F (Φ) F for Φ p H(G p,k p ) L(F ; s) := p L p(λ F ; s) F L 5.3 Hecke compatibility 5.2 Jacobi f S k,1 (Γ S ) ph S,p (1) λ f (φ 0,p )=1 I(f) K p λ f (φ 0,p ) 1 λ f (φ 0,p )= p p 1 I(f)(gu) du =0 K p (2) λ f (φ 0,p )=1 (for p) I(f) S k (Γ ) p< H p (G p,k p ) L(I(f); s) =L(f; s) m ζ(s + j m/2) j=0 5.1 f S k,1 (Γ S ) H S,p I(f) H(G p,k p ) (2) 6. I I S maximal Maass I : S k,1 (Γ S ) S k (Γ ) Petersson adjoint I f,i (F ) k,1 = I(f),F k f S k,1 (Γ S ), F S k (Γ ). 6.1 k>2m +4 f S k,1 (Γ S ) Hecke I I(f) =C S,k L(f;1+m/2) f [(m+1)/2] C S,k =(dets) (m+1)/2 { j=1 B 2j (4π) k Γ(k) 2 m/2 π 1 m/2 m :even 2 (m+1)/2 Γ((m +3)/2) 1 m :odd } { 4 1 Γ 2 1 Γ } I(f) G A η V 1,Q (adelic) Fourier F η (g) := F (n(x)g) χ( Q(η, x)) dx V 1,Q \V!,A 15

220 210 η = a ( α L S Sα 1, iη D T := 1 t αs 2a T H 1 T 1 := compact U p, U 1,p 1 T 1 ) maximal H p H p,h 1,p U p = {h H p GL m+1 (Z p ) (h 1)T 1 M m+1 (Z p )} U 1,p = {h H 1,p GL m+3 (Z p ) (h 1)T 1 1 M m+3 (Z p )} H1, 0 X := {(x, r) Rm+1 R r>0} h 1 r + T [x]/2 x = r + T [x ] x J H1 (h 1, (x, r)) 1 1 (x, r) h 1 (x, r) := (x,r ) (1, 0) X U1, SO(m +1) H A UA := H0 p U p S(U A ) ϕ S(U A ) H 1 Eisenstein E(h 1,ϕ; s) = ϕ(β(γh 1 )) α(γh 1 ) s+(m+1)/2 A γ P 1,Q \H 1,Q α(h 1 ) h 1 = β(h 1 ) u(h 1 ) P 1,A U1, U1,p α(h 1 ) 1 p 6.2 F S k ( p K p ), ϕ S(U A ) F (h 1 g η ) E(h 1,ϕ; s 1/2) dh 1 H 1,Q \H 1,A { t } = F η ( h g η ) ϕ(h) dh t s (m+2)/2 d t H Q \H A Q A t 1 g η G 0 1, g η Z 0 = iη(q 1 [η]/2) 1/ H A 1 E(h 1, 1; s) s =(m +1)/2 1 Res E(h 1, 1; s) = (2π)(m+1)/2 Γ((m +1)/2) s=1+m/2 (det T ) 1/2 Γ(m +1) ζ(m +1) ζ(m +2) L(χ T ;(m +1)/2) L(χ T ;(m +3)/2) 16 Res s=1 ζ(s) ζ(m +1) m :even m :odd p< B T,p (p (m+1)/2 ) B T,p (p (m+3)/2 )

221 211 h H Q \H A m+1 vol(h Q \H A )=vol(ha)2 1 m π (m+1)(m+2)/2 { B T,p (p (m+1)/2 ) p< j=1 [m/2] Γ(j/2)(det T ) m/2 j=1 1 m :even L(χ T ;(m +1)/2) m :odd ζ(2j) 6.1 F = I(f) F η H A 3 [1] K. Doi and H. Naganuma : On the functional equation of certain Dirichlet series, Invent. math. 9 (1969), [2] :theta [3] H, Naganuma : On the coincidence of two Dirichlet series associated with cusp forms of Hecke s Neben -type and Hilbert modular forms over a real quadratic field, J. Math.Soc.Japan25 (1973), [4] S. Niwa : Modular forms of half integral weight and the integral of certain thetafunctions, Nagoya Math. J. 56 (1974), [5] T. Oda : On modular forms associated with indefinite quadratic forms of signature (2,n 2), Math. Ann. 231 (1977), [6] T. Shintani : On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), [7] S. Sugano : Jacobi forms and the theta lifting, Comment. Math. Univ. St. Pauli 44 (1995), [8] D. Zagier : Modular forms associated to real quadratic fields, Invent. math. 30 (1975), [9] D. Zagier : Sur la conjecture de Saito-Kurokawa (d apres H Maass), Seminaire Delange- Pisot-Poitou , Progr. Math. Vol. 5 (1980), , Birkhauser. 17

222 212 Borcherds product ( ) [3, 6, 25] [22, 26] H := {τ C Im τ > 0} { ( ) a b SL(2, R) := M = M(2, R) c d ad bc = 1 } 1

223 213 SL(2, R) H SL(2, R) H (M, τ) M τ H M τ := aτ + b cτ + d ( ( ) ) a b M = SL(2, R) c d τ H, E 2 τ = τ M, M SL(2, R), τ H, M M τ = (MM ) τ E 2 ( E 2 ) τ = τ Hol(H) k Z SL(2, R) Hol(H) Hol(H) SL(2, R) (f, M) f k M Hol(H) (f k M)(τ) := (cτ + d) k f(m τ ) ( ( ) ) a b M = SL(2, R) c d f Hol(H), f k E 2 = f M, M SL(2, R), f Hol(H), (f k M) k M = f k (MM ) k SL(2, R) SL(2, Z) := SL(2, R) M(2, Z) f Hol(H) k M SL(2, Z), f k M = f f τ 1 f(τ) = a f (n)q n n= q := e(τ) := exp(2π 1τ) f Hol(H) k f 2

224 214 f k n < 0 a f (n) = 0 f n 0 a f (n) = 0 f f N n < N a f (n) = 0 f k C- M k C- M cusp k E 2 SL(2, Z) k M k = M cusp k = {0} e k (τ) = 1 2 (c,d)=1 1 (cτ + d) k k 2 k k (c, d) ( c, d) 0 e k (τ) = 1 + C k n=1 σ k 1 (n)q n M k σ k 1 (n) n k 1 σ k 1 (n) := d k 1 0<d n C k B k C k := 2k B k C 4 = 240, C 6 = 504, C 8 = 480, C 10 = 264, C 12 = ,... (τ) := q (1 q n ) 24 M cusp 12 n=1 3

225 215 H (τ) 12 [5] 1. f f f(τ) = a f (n)q n n=n a f (0) 0 Proof. ρ := , i := 1 f a f (0) = 1+ρ ρ f(τ)dτ ρ i 1 + ρ i f τ 1 τ a f (0) = 0 SL(2, Z) 2. k 0 0 k f M k e 4, e 6 f(τ) = 4a+6b=k c a,b e 4 (τ) a e 6 (τ) b (c a,b C) M Z M Z := k Z M k = C[e 4, e 6 ] 4

226 216 (τ) = e 4(τ) 3 e 6 (τ) M k f f M cusp k+12 [13, 31] 2.2 g H g := {Z M(g, C) Z = t Z, Im Z > 0} g Im Z > 0 Z { ( ) A B Sp(g, R) := M = M(2g, R) t MJ C D g M = J g := ( O g E g )} E g g O g g E g g Sp(g, R) H g Sp(g, R) H g (M, Z) M Z H g O g ( ( ) ) M Z := (AZ + B)(CZ + D) 1 A B M = Sp(g, R) C D ( E 2g ) Z = Z H g Hol(H g ) k Z Sp(g, R) Hol(H g ) Hol(H g ) Sp(g, R) (F, M) F k M Hol(H g ) (F k M)(Z) := det(cz +D) k F (M Z ) ( ( ) ) A B M = Sp(g, R) C D 5

227 217 kg g = 1 H 1 = H Sp(1, R) = SL(2, R) Sp(g, R) Sp(g, Z) := Sp(g, R) M(2g, Z) g = 2 F Hol(H g ) g k M Sp(g, Z), F k M = F g = 1 F F Z 1 F (Z) = T = t T a F (T )e(tr(t Z)) T g g 2 3. g F k F F F (Z) = T = t T a F (T )e(tr(t Z)) T 0 a F (T ) = 0 T 0 T T > 0 a F (T ) = 0 F F E 2g Sp(2, Z) kg 0 Borcherds g = 2 g = 2 g = 2 k C- M k 6

228 218 C- M cusp k H 2 Z ( ) τ z Z = z ω F (Z) F (τ, z, ω) 4. F M k F F F (Z) = n,l,m Z c(n, l, m) e(nτ + lz + mω) 4nm l 2 0 n 0 c(n, l, m) = [24] 5. k 0 0 k F M k E 4, E 6, 10, 12 4, 6, 10, 12 F (Z) = 4a+6b+10c+12d=k c a,b,c,d E 4 (Z) a E 6 (Z) b 10 (Z) c 12 (Z) d (c a,b,c,d C) k k M Z M 2Z := k 2Z M k = C[E 4, E 6, 10, 12 ] M Z := k Z M k = M 2Z 35 M 2Z [13, 16, 27] 7

229 F M k F (Z) = φ m (τ, z)e(mω) m=0 φ m m m Z i m : Hol(H C) Hol(H 2 ) (i m φ)(z) := φ(τ, z)e(mω) Sp(2, R) J := {M Sp(2, R) φ, ψ s.t. (i 1 φ) 0 M = i 1 ψ} Sp(2, R) J Sp(2, R) M Sp(2, R) J m Z, k Z, φ, ψ s.t. (i m φ) k M = i m ψ ψ φ k,m M k, m Z Sp(2, R) J Hol(H C) Sp(2, R) J T (u) := u ( u R ) a 0 b 0 C(a, b, c, d) := c 0 d 0 ( a, b, c, d R, ad bc = 1 ) y U(x, y) := x 1 y x ( x, y R )

230 220 Hol(H C) (φ k,m T (u)) (τ, z) =e(mu)φ(τ, z) ( mcz (φ k,m C(a, b, c, d)) (τ, z) =(cτ + d) k 2 e cτ + d ) ( aτ + b φ cτ + d, (φ k,m U(x, y)) (τ, z) =e ( m(x 2 τ + 2xz + xy) ) φ(τ, z + xτ + y) ) z cτ + d 6. Sp(2, R) J a, b, c, x, y, u R C(a, b, c)u(x, y)t (u) S := S Sp(2, R) J 7. Sp(2, R) Sp(2, R) J S S Hol(H 2 ) (F k S) (τ, z, ω) = ( 1) k F (ω, z, τ) Sp(2, Z) J := Sp(2, R) J M(4, Z) 8. Sp(2, Z) J a, b, c, x, y, u Z C(a, b, c)u(x, y)t (u) 9. Sp(2, Z) Sp(2, Z) J S φ Hol(H C) k m M Sp(2, Z) J, φ k,m M = φ 9

231 221 ( ) ( ) (( ) ) mcz φ(τ, z) =(cτ + d) k 2 aτ + b e φ cτ + d cτ + d, z a b SL(2, Z) cτ + d c d φ(τ, z) =e ( m(x 2 τ + 2xz) ) φ(τ, z + xτ + y) ( x, y Z ) m φ Hol(H C) k m m < 0 φ = 0 m = 0 φ τ z Proof. τ φ z 1 2π 1 φz (τ, z) dz = 2m φ(τ, z) m 0 m = 0 φ z 0 z m 0 φ Hol(H C) k m φ τ, z 1 φ(τ, z) = c(n, l)q n ζ l n,l Z q := e(τ), ζ := e(z) m > 0 c(n, l) 4nm l 2 l mod 2m c(n, l) = ( 1) k c(n, l) m = 0 l = 0 c(n, l) = 0 φ Hol(H C) k m φ φ k m 4nm l 2 < 0 c(n, l) = 0 10

232 222 4nm l 2 0 c(n, l) = 0 φ n < 0 c(n, l) = 0 φ N n < N c(n, l) = 0 φ m = 0 k 0 φ(τ, z) z τ k k 0 k k 0 0 k m = 0 k m C- J k,m C- J cusp k,m k m C- J weak k,m C- J wh k,m J cusp k,m J k,m J weak k,m Jwh k,m φ 2,1 (τ, z) := ( ζ 2 + ζ 1) φ 1,2 (τ, z) := ( ζ ζ 1) n=1 n=1 (1 q n ζ) 2 (1 q n ) 4 ( 1 q n ζ 1) 2 J weak 2,1 ( 1 q n ζ 2) (1 q n ) 2 ( 1 q n ζ 2) J weak 1, φ 10,1 := (τ)φ 2,1 (τ, z) J cusp 10, (τ, z) := 1 z 2 + w (Z+Zτ)\{0} { 1 (z w) 2 1 } w 2 = 1 z (2k 1)e 2k (τ)z 2k 2 k=2 11

233 223 z = Z + Zτ (τ, z) ζ ζ 1 ( ) 2 = 2π 1 ζ 2 + ζ a ( ζ a 2 + ζ a) q n n=1 a n φ 0,1 := 12 (τ, z)φ 2,1(τ, z) ( 2π 1 ) 2 J weak 0, m 0 m 2m 0 k φ J weak k,m e 4, e 6, φ 2,1, φ 0,1 4, 6, 2, 0 0, 0, 1, 1 φ(τ, z) = c a,b,c,d e 4 (τ) a e 6 (τ) b φ 2,1 (τ, z) c φ 0,1 (τ, z) d (c a,b,c,d C) 4a+6b 2c=k, c+d=m k k + 1 φ 1,2 J weak 2Z,Z := k 2Z, m Z J weak Z,Z J weak k,m = M Z [φ 2,1, φ 0,1 ] = C[e 4, e 6, φ 2,1, φ 0,1 ] k Z, m Z J weak k = J weak 2Z,Z φ 1,2 J weak 2Z,Z [15]

234 t N φ J k,m (φ k,m V (t)) (τ, z) := t k 1 d 1 ad=t, a>0 b=0 ( ) aτ + b d k φ, az J k,mt d φ J cusp k,m φ k,mv (t) J cusp k,mt 13. k φ J cusp k,1 (ML(φ)) (Z) := m=1 (φ k,1 V (m)) (τ, z)p m M cusp k p := e(ω) ML(φ 10,1 ) = 10 ML(φ 12,1 ) = 12 (φ 10,1 (τ, z) := (τ)φ 2,1 (τ, z) J cusp 10,1 ) (φ 12,1 (τ, z) := (τ)φ 0,1 (τ, z) J cusp 12,1 ) E 4, E 6 φ 4,1 := e 4(τ)φ 0,1 (τ, z) e 6 (τ)φ 2,1 (τ, z) 12 φ 6,1 := e 6(τ)φ 0,1 (τ, z) e 4 (τ) 2 φ 2,1 (τ, z) 12 J 4,1 J 6, φ φ(τ, z) = n,l Z c(n, l)q n ζ l J cusp k,1 13

235 225 (ML(φ)) (Z) := (φ k,1 V (m)) (τ, z)e(mω) m=1 = m d 1 k 1 m=1 m=1 ad=m b=0 = m d 1 k 1 d k m=1 ad=m b=0 ad=m ( ) aτ + b d k φ, az e(mω) d n,l Z ( c(n, l)e n aτ + b ) + alz + mω d = m k 1 d k+1 c(dn, l)e (naτ + alz + mω) = = m=1 ad=m a=1 n,l,m Z = n,l,m Z a (n,l,m) n,l Z n,l Z a ( nm ) k 1 c a, l e (naτ + alz + mω) a k 1 c(nm, l)e (naτ + alz + amω) ( nm a k 1 c a 2, l ) e (nτ + lz + mω) a ML(φ) S 9 ML(φ) Sp(2, Z) k φ k,1 V (0) [9] φ J wh k,1 φ k,1 V (0) 3.2 k = 0 J 0,1 = {0} ML(0) = 0 φ(τ, z) = c(n, l)q n ζ l J cusp 0,1 n,l Z 14

236 226 (ML(φ)) (Z) = = a 1 c(nm, l)e (naτ + alz + amω) a=1 n,l,m Z = n,l,m Z = n,l,m Z exp( ML(φ)) = c(nm, l) a 1 e (nτ + lz + mω) a a=1 c(nm, l) log (1 e (nτ + lz + mω)) n,l,m Z ( 1 q n ζ l p m) c(nm,l) S n,l,m Z 0 ( 1 q n ζ l p m) c(nm,l) J cusp 0,1 = {0} exp(0) = 1 φ φ 0 φ φ c(n, l) n > 0 m=1 τ ω S- φ n = 0 φ k,1 V (0) φ n < 0 S- exp( ML(φ)) φ(τ, z) := 2φ 0,1 (τ, z) J weak 0, 1 15

237 227 φ(τ, z) = c(n, l)q n ζ l n,l Z = ( 2ζ ζ 1) + ( 20ζ 2 128ζ ζ ζ 2) q + BP(φ)(Z) := qζ 1 ( p 1 q n ζ l p m) c(nm,l) (n,l,m)>0 (n, l, m) > 0 m N, n, l Z m = 0, n N, l Z m = n = 0, l N m 1 exp( ML(φ)) qζ 1 p ( 1 q n ζ l) c(0,l) n,l =q ( ζ 2 + ζ 1) p (1 q n ζ) 2 (1 q n ) 20 ( 1 q n ζ 1) 2 n=1 = (τ)φ 2,1 (τ, z)p 10 1 BP(φ)(Z) := q ( ζ 2 + ζ 1) p ( 1 q n ζ l p m) c(nm,l) n,m 0 l Z BP(φ) S- BP(φ) H 2 BP(φ) ML(φ 10,1 ) = BP(2φ 0,1 ) (= 10 ) φ = 2φ 0,1 φ J wh k,0 φ(τ, z) = c(n, l)q n ζ l J wh k,0 n,l Z 16

238 228 BP(φ)(Z) := q a ζ b p c ( 1 q n ζ l p m) c(nm,l) (n,l,m)>0 (n, l, m) > 0 a := 1 c(0, l), b := 1 c(0, l)l, c := 1 c(0, l)l l Z l>0 m 1 exp( ML(φ)) q a ζ b p c ( 1 q n ζ l) c(0,l) (n,l)>0 (n, l) > 0 n N, l Z n = 0, l N l>0 14. a, b Z 1 2c(0, 0) Z q a ζ b (n,l)>0 ( 1 q n ζ l) c(0,l) 1 2c(0, 0) c Proof. BP(φ) Sp(2, Z) J BP(φ) S- 15. Proof. a c n>0, m<0, l Z n>0, m<0, l Z nc(nm, l) = nc(nm, l) = 0 σ 1 (n)c( n, l) n=1 l Z (τ) 24(2π 1) (τ) = 1 24 σ 1 (n)q n n=1 17

239 229 a n>0, m<0, l Z nc(nm, l) (τ)φ(τ, 0) 24(2π 1) (τ) τ c φ zz (τ, 0) 4(2π 1) 2 (τ)φ(τ, 0) 24(2π 1) (τ) φ zz(τ, 0) 4(2π 1) n>0, m<0, l Z c(nm, l) Z BP(φ) S- Proof. BP(φ) H 2 BP(φ) 1 2c(0, 0) S- Borcherds Sp(2, Z) IV 18

240 IV s (2, s + 2) S S 0 M(s, Z) 1 S = S 1 S 1, S 1 = 1 S 0 1 O(S, R) := { M M(s + 4, R) t MSM = S } H S := { w C s+4 S[w] := t wsw = 0, S{w} := t wsw > 0 } w Mw O(S, R) H S O(S, R) G H S t (1, 1, 0,..., 0, 1, 1) H 0 S O(S, R) H0 S H0 S G G O(S, R) S IV ω H S := Z = z C s+2 ω, τ C, z C s, S 1 [Im Z] > 0, Im τ > 0 τ HS 0 P CHS 0 H S 1 2 S 1[Z] H S Z Z P C HS 0 1 G H S g 0,0 g 0,1... g 0,s+3 g 1,0 g 1,1... g 0,s+3 M = G g s+3,0 g s+3,1... g s+3,s+3 19

241 231 ω Z = z = τ z 1 z 2. z s+1 z s+2 H S z = z 2. z s+1 J i (M, Z) := 1 2 g s+2 i,0s 1 [Z] + g i,j z j + g i,s+3 j=1 G H S G H S (M, Z) M Z := ( ) s+2 Ji (M, Z) H S J s+3 (M, Z) i=1 J(M, Z) := J s+3 (M, Z) Z H S, J(E s+4, Z) = 1 M, M G, Z H S, J(MM, Z) = J(M, M Z )J(M, Z) IV H S Hol(H S ) k Z G Hol(H s ) Hol(H S ) G (F, M) F k M Hol(H S ) (F k M)(Z) := J(M, Z) k F (M Z ) S 8 s Γ := G O(S, Z) F Hol(H S ) k M Γ, F k M = F IV 20

242 F k F F F (Z) = n,m Z,l Z s a(n, l, m)e(nτ + t ls 0 z + mω) 2nm S 0 [l] 0 n 0 a(n, l, m) = 0 2nm S 0 [l] > 0 n > 0 a(n, l, m) = 0 F IV S 0 = (2) S s = IV k F F (Z) = φ m (τ, z)e(mω) m=0 φ m m m Z i m : Hol(H C s ) Hol(H S ) (i m φ)(z) := φ(τ, z)e(mω) G J := {M G φ, ψ s.t. (i 1 φ) 0 M = i 1 ψ} G J G J := G J G G G M G J m Z, k Z, φ, ψ s.t. (i m φ) k M = i m ψ ψ φ k,m M k, m Z G J Hol(H C s ) 21

243 233 IV 1 G F := M = M 1 G t M 1 S 1 M 1 = S G GJ G F S(s) := 0 0 E s 0 0 G F S(s) G J G S(s) Hol(H 2 ) (F k S(s)) (τ, z, ω) = F (ω, z, τ) Γ = G O(S, Z) Γ Γ J := Γ G Γ F := Γ G F 19. Γ Γ J Γ F Γ G = Γ Γ Γ 2 Γ/Γ {E s+4, S(s)} φ Hol(H C s ) k m M Γ J, φ k,m M = φ m 0 22

244 φ Hol(H C s ) k m m < 0 φ = 0 m = 0 φ τ z S 0 θ S (τ, z) = ( ) 1 e 2 S 0[l] + t ls 0 z l Z s s k 1 k s H 2 φ(τ, z) = f(τ)θ S (τ, z) f(τ) k 1 φ φ(τ, z) = c(n, l)e(nτ + t ls 0 z) n Z,l Z s c(n, l) n 1 ( 2 S 0[l] c n 1 ) 2 S 0[l] f(τ) := n c(n )q n k s 2 H 4.3 IV IV 23

245 k 1 φ φ(τ, z) = n Z,l Z s c ( n 1 ) 2 S 0[l] e(nτ + t ls 0 z) c(n ) n 0 c(n ) = 0 (ML(φ)) (Z) := n,m Z,l Z s a (n,l,m) ( ( 1 a k 1 c a 2 nm 1 )) 2 S 0[l] e ( nτ + t ls 0 z + mω ) H S k IV 0 1 φ φ(τ, z) = ( n 1 ) 2 S 0[l] e(nτ + t ls 0 z) n Z,l Z s c c(n ) n c(n ) = 0 n < 0 c(n ) Z a := 1 ( c 1 ) 24 2 S 0[l], b := 1 ( c 1 ) 2 2 S 0[l] l, l Z s l>0 c := 1 ( c 1 ) 4 2 S 0[l] S 0 [l] l>0 a, c Z, b Z s l > 0 l Z s BP (φ) BP(φ)(Z) :=e(aτ t bs 0 z + cω) ( 1 e(nτ + t ls 0 z + mω) ) c(nm 1 2 S 0[l]) (n,l,m)>0 c(0) (n, l, m) > 0 2 m N, n Z, l Z s m = 0, n N, l Z s m = n = 0, l > 0 24

246 236 [9] 23. BP(φ) S 1 [Im Z] BP(φ) H S BP(φ) H S c(0) 2 c(n ) (n < 0) c(n ) [9] log H S c(n ) c(n ) c(n) 2π 1 c m>0 c>0 0 a<c,0 d<c,c (ad 1) ( ) ( ) an md 4π mn (m c( m)e I 1+ s 2 c c n ) s 4 I ϵ O(exp(ϵ n)) [9] 25

247 [22] [12] log H S [33] s H f 2 dx dy f(τ)θ S (τ, Z)y y 2 (τ = x + y 1) F Z H SL(2, Z) F u 1+s dx dy f(τ)θ S (τ, Z)y F u y 2 s u 0 s = 0 [12] 5.3 φ m (τ, z)e(mω) m Γ F 2, 3, 4 φ 26

248 238 6 [23] [8] [1, 14] [10, 29] [21, 28] 7 27

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252 242 Siegel Eisenstein Fourier 1 Siegel Eisenstein Fourier Eisenstein Siegel Eisenstein Fourier Ikeda lift Fourier Eisenstein Fourier 1999 [Kat] Maass [Ma] Fourier 2 Siegel Fourier Siegel [Kl] H n = {Z M g (C) t Z = Z, Im(Z) 0}, ( ) Γ = Γ n = Sp(n, Z) = {γ GL(2n, Z) γj t 0 1n n γ = J n }, J n =, 1 n 0 {( ) } A B = A C D t B = B t A, C t D = D t C, A t D B t C = 1 n, Sym n (Z) = {N M n (Z) N = t N}. ( ) A B H n f γ = Γ C D f k γ(z) = det(cz + D) k f((az + B)(CZ + D) 1 ) Γ k Siegel { } M k (Γ ) = f : H n C f k γ = γ, γ Γ, n = 1 f cusp 1

253 243 f M k (Γ ) f Z Z + N, N Sym n (Z) Z Fourier Sym n (Z) dual lattice S n = {A Sym n (Q) Tr(AN) Z, N Sym n (Z)} = {A = (a ij ) Sym n (Q) a ii Z, a ij 1 Z (i j)} 2 S n f M k(γ ) f(z) = A S n c(a) exp(2πi Tr(AZ)) A 0 c(a) = 0 (Koecher ) f M k (Γ ) f(z) = A S n A 0 c(a) exp(2πi Tr(AZ)) Fourier Fourier U GL n (Z) ( t U 0 0 U 1 ) Sp(n, Z) det(u) k C( t UAU) = C(A) 3 Siegel Eisenstein Fourier 3.1 Eisenstein Γ n = {( ) } A B Γ n 0 D k E n k (Z) = ( A B C D ) Γ n \Γ n det(cz + D) k k well-defined k > n + 1 (k k n + 2) M k (Γ ) [Kl] 5 Γ \H n 2

254 Symmetric co-prime pair E n k (Z) M k(γ ) Fourier Γ \Γ 1 (1) C, D M n (Z) ( ) Γ C D { (i) C t D (ii) M, N M n (Z) CM + DN = 1 n (i) symmetric (ii) co-prime (i),(ii) (C, D) M n (Z) 2 symmetric co-prime pair M n symmetric co-prime pair (2) Γ \Γ GL n (Z)\M n, ( ) (C, D) C D GL n (Z) U M n (C, D) (UC, UD) M n ) (1) CM +DN = 1 n M, N M n (Z) A = t N + t MNC, B = ( t M + t ) MND A t B B t A = 0, A t D B t C = 1 n A B Γ n C D (2) symmetirc co-prime pair (C, D) C 0 r n M r n = {(C, D) M n rank C = r} M 0 n = {(0, U) U GL n (Z)} GL n (Z) M r n GL n(z) Eisenstein E n,0 k E n k (Z) = E n,r k (Z) = n r=0 (C,D) GL n (Z)\M r n E n,r k (Z), det(cz + D) k (Z) = 1 En,r(Z) k 3

255 r < n GL n (Z)\M r n Λ n,r = {Q M n,r (Z) S M n,n r (Z) s.t. (Q, S) GL n (Z)} Q Λ n,r Q = (Q, ) GL n (Z) 2 GL n (Z)\M r n {(( ) ( ) ) C 0 t D 0 Q, Q n r (C, D ) GL r (Z)\M r r Q Λ n,r /GL r (Z) } E n,r k (Z) = Q = Q (C,D ) (C,D ) det (( ) ( ) ) C 0 t D k 0 QZ + Q n r ( ( ) C 0 det 0 0 ( ) t D 0 QZQ n r } {{ } ( ) ) k ( ) C ( ) = W + D, W = ( t QZ Q (r, r)-block) 0 1 n r E n,r k (Z) = Q (C,D ) GL r (Z)\M r r det(c Z[Q] + D ) k, (Z[Q] = t QZQ H r ) (3.1) E n,r k z H r Fourier Eisenstein Ek r (Z) Fourier E r k(z) = E n,r k (Z) = B S r Q Λ n,r /GL r (Z) B Sr C (B) exp(2πi Tr(Bz)) = Q,B C (B) exp(2πi Tr(QB t QZ)) C (B) exp(2πi Tr(BZ[Q])) rank QB t Q < r A Sn Fourier ( 1) E n,n k (Z) Fourier C(A) A Fourier 4

256 E n,n k (Z) Fourier C E n,r k (Z) = (C,D) M n n = (C,D) det(cz + D) k det C k det(z + C 1 D) k symmetric C 1 D 3 (C, D) GL n (Z)\M n n 1:1 Sym n (Q), (C, D) C 1 D ) T Sym n (Q) U, V GL n (Z) UV T = ν 1 /δ 1... ν n /δ n, (ν i, δ i ) = 1, δ i δ i+1, δ i > 0 T 1 δ... U 1, ν 1... V GL n (Z)\M n n δ n ν n GL n (Z)\M n n Sym n (Q) GL n (Z)\M n n (C, D), (C 1, D 1 ) M n n C 1 D = C 1 1 D 1 (C, D) (C 1, D 1 ) GL n (Z)- U = C 1 C 1 UC = C 1 M n (Z), UD = D 1 M n (Z) (C, D) co-prime U M n (Z) U 1 = CC 1 1 M n (Z) U GL n (Z) U(C, D) = (C 1, D 1 ) T Sym n (Q) δ(t ) = i δ i (C, D) T δ(t ) = det C S Sym n (Z) δ(t + S) = δ(t ) (C, D) T ( ) ( ) ( ) 1n S = Sp(n, Z) C D 0 1 n C CS + D 5

257 247 1 (C, CS + D) M n n Mn n (C, D + CS) T + S E n,n k = = S Sym n (Z) T Sym n (Q) T Sym n (Q) mod Sym n (Z) δ(t ) k det(z + T ) k δ(t ) k A S n S Sym n (Z) det(z + T + S) k (3.2) Z = X + iy H n Fourier det(z + S) k = ξ n (Y, A, k) exp(2πi Tr(AX)) (3.3) Fourier ξ n (Y, A, k) ξ n (Y, A, k) = det(x + iy ) k exp( 2πi Tr(AX)) dx Sym n (R) (3.3) (k > n) ξ n (Y, A, k) = ξ n (A, k) exp( 2π Tr Y ) X X + T (3.2) E (n) k (Z) = δ(t ) ξ n (A, k) exp(2πi Tr(AT )) exp(2πi Tr(AZ)) T A Sn = ξ n (A, k)b n (A, k) exp(2πi Tr(Z)) (3.4) A S n b n (A, k) = T Sym n (Q) mod Sym n (Z) δ(t ) k exp(2πi Tr(AT )) (3.4) ξ n b n 3.5 ξ n (Y, A, k) n = 1 n = 2 Kaufold ([Kau]) n Siegel Shimura([Sh1]) m 1 Γ m (s) = π m(m 1)/4 6 j=0 Γ (s j/2)

258 248 1 ([Si, (111)], [Sh1, (3,15),(4.7K),(4,10)]) A S n 2 n(n 1) 2 ( 2πi) nk n+1 k (det A) 2 A 0 ξ n (A, k) = Γ n (k) 0 ξ n (A, k) A det A [Sh1] A α, β C (re(α + β) 0) ξ n (Y, A; α, β) = det(x + iy ) α det(x iy ) β exp( 2πi Tr(AX)) dx Sym n (R) ( ) (α, β) C 2 A (p+, q ) ξ n (Y, A; α, β) = Γ n p q(α + β n+1 2 ) (α, β ) Γ n q (α)γ n q (β) k > n + 1 A Γ n q (β) ξ n (Y, A, k) = ξ n (Y, A; k, 0) = Siegel s C b n (A, s) = T Sym n (Q) mod Sym n (Z) δ(t ) s exp(2πi Tr(AT )) Siegel ( singular ) T Sym n (Q) T = T p1 + T p2 + + T pr, T pi Sym n (Q), δ(t pi ) = p e i i, (p i ) Sym n (Q)/ Sym n (Z) mt Sym n (Z) m Z 1/m = i q i/p f i i T pi = (q i m/p f i i )T δ(t ) = i δ(t p i ) b n (A, s) Euler b n (A, s) = b p n(a, s) p b p n(a, s) = T Sym n (Q p ) mod Sym n (Z p ) δ(t ) s e p (Tr(AT )) 7

259 249 e p e p : Q p /Z p m 1 p m Z/Z e 2πi( ) C T Sym n (Q p ) δ(t ) p b p n(a, s) n = 2 Kaufhold ([Kau]) n Siegel, Feito, Shimura, Kitaoka 1999 Katsurada ([Kat]) b p n(a, s) p s Q- Shimura ([Sh2]) Langlands ([La]) Siegel Rieman Euler Euler Kitaoka n Q( ( 1) n det(2a))/q D A χ A 2 χ A p χ A (p) = ( DA ( 1) n det(2a) = D A f 2 A f A N p ) 2 ([Ki, Theorem 2]) A Sym n (Q) F p (A, T ) Z[T ] n/2 (1 p s ) (1 p 2j 2s )(1 χ A (p)p s+n/2 ) 1 n: even j=1 b n (A, s) = n 1 F p (A, p s ) 2 (1 p s ) (1 p 2j 2s ) n: odd j=1 F p (A, T ) p 1 F p (A, T ) Katsurada [Kat] n 1 ( ) n > k + 1 Eisenstein Fourier E n,r k (Z) = C(A) exp(2πi Tr(AZ)) A Sn,A 0 8

260 250 (1) A 0 C(A) = ξ n (A, k) p b p n(a, k) Euler 1 2 (2) rank A = r < n A = QA t Q Q Λ n,r S r A 0 C(A) = ξ r (A, k) p b p r(a, k) Siegel Siegl Katsurada[Kat] n n 3 n F p (A, p n 1 T 1 ) = (p n+1 2 T ) f A F p (A, T ) Ikeda lift Siegel S Sym m (Z) T Sym n (Z) {X M m,n (Z/p l Z) t XSX T mod p l } A p l(s, T ) α p (S, T ) = lim l p l(n(n+1)/2 mn) A p l(a, T ) l stable ( ) 0 1k 4 H k = k 0 S n A 0 b p n(a, k) = α p (H k, A) 9

261 251 Siegel Z p 2 p Z 2 theta Eisenstein theta Siegel-Weil Arakawa [Ar] 4.2 Eisenstein Fourier n = 1 Fourier Siegel m Z >0 b p 1 (m, s) m = p t m, (p, m ) = 1 b p 1 (m, s) = δ(r) s e p (mr) r Q p mod Z p ( 2πim = 1 + p ls ) u exp p l t. l=1 u (Z/p l ) u = u 2 p + u 1, u 2 Z/p l 1, u 1 (Z/p) b p 1 (m, s) = 1 + l=1 p ls u 2 Z/p l 1 exp 2πi ( m ) u 2 ( m ) u 1 p l t 1 exp 2πi p l t. u 1 (Z/p) l t + 2 t+1 b p 1 (m, s) = 1 + ( m p ls+l 1 ) u 1 exp 2πi p l t l=1 u 1 (Z/p) u 1 (Z/p) exp 2πi ( m ) { u 1 p 1 p l t = 1 l t, l = t + 1 t S p (n, s) = 1 + p l(1 s) 1 (p 1) p (1+t)(1 s) 1 l=1 = (1 p s ) t l=0 10 p (1 s)l

262 252 F p (m, T ) = ord p m l=0 (pt ) l b 1 (m, k) = p b p 1 1 (m, k) = ζ(k) p ord p m l=0 p (k 1) k ξ 1 (m, k) = (2πi)k (k 1)! mk 1 C(m) = (2πi)k σ k 1 (m), σ l (m) = d l ζ(k)(k 1)! d m n = ξ 2 (A, k) = b 2 (A, k) = L(k 1, χ A) ζ(k)ζ(2k 2) (2π)k det(a) k 3/2 2 πγ(k)γ(k 1/2) F 2 (A, p k ) ( ) a1 a 2 F p (A, T ) [Kau] A = a 2 a 3 α 1 = ord p (gcd(a 1, 2a 2, a 3 )), α = ord p f A { α 1 α l F p (A, T ) = (p 2 T ) l (p 3 T 2 ) m χ A (p)pt l=0 m=0 p α l 1 m=0 (p 3 T 2 ) m } 11

263 253 [Ar] T. Arakawa, 2 I(Siegel Eisenstein ), 1 [Kat] H. Katsurada, An explicit formula for Siegel series, Amer. J. Math. 121 (1999), [Kau] G. Kaufhold Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades, Math. Ann. 137, 1959, [Ki] Y. Kitaoka, Dirichlet series in the theory of quadratic forms, Nagoya Math. J. 92 (1984), [Kl] H. Klingen, Introductory lectures on Siegel modular forms, Camblridge studies in advanced math. 20, 1990 [La] R. P. Langlands On the functional equations satisfied by Eisenstein series, Lecture notes in Math. 544, Springer, [Ma] H. Maass Siegel s modular forms and Dirichlet series, Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin-New York, 1971 [Sh1] G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann. 260 (1982), no. 3, [Sh2] G. Shimura, On Eisenstein series, Duke Math. J., 50 (1983), [Si] C.L. Siegel, Über die analytishe Theorie der quadratischen Formen I, Ann. Math. 36 (1935),

264 254 Functoriality Principle summer school 90. Functoriality principle summer school,,.,.,.. F G F K, G(K) G K. G R group scheme, S R-algebra, G(S) G S-valued points. global field,. F global field. F A, F A F,. G F. G G A. F A F -algebra G A = G(F A ). G A automorphic representation A(G A ). reductive algebraic group reductive group. 0. Motivation E Q elliptic curve. Hecke eigenform f S 2 (Γ 0 (N)) (0.1) L(s, E) = L(s, f). N E conductor. functoriality principle.. (1) E K. Deuring K A /K Hecke ψ L(s, E) = L(s, ψ). ψ GL(1, K A ) automorphic form, ψ f GL(1, K A ) automorphic form GL(2, K A ) automorphic form. L modular form Hecke, endoscopic lift. 1

265 255 (2) F, E F elliptic curve. naive Γ = SL(2, O F ) weight (2, 2) Hilbert modular form f L(s, E) = L(s, f). 1 Q elliptic curve E 0 E E 0 F elliptic curve. E E 0 F base change E = E 0 Q F. E 0 elliptic modular form f 0 Hilbert modular form f, L(s, f) = L(s, f 0 )L(s, f 0 χ). χ F Dirichlet. f 0 elliptic curve elliptic modular form f 0 Hilbert modular form f, lift, base change lift. (3) D F quaternion algebra. D A automorphic form GL(2, F A ) automorphic form Hecke. A(D A ) A(GL(2, F A)). Eichler-Shimizu-Jacquet-Langlands, functoriality. (4) (0.1), f GL(2, Q A ) automorphic representation π = v π v. v = p, E p potential good reduction π p special (Langlands-Deligne-Carayol).. F ρ λ : Gal(F /F ) GL(d, E λ ) λ-adic representation. ρ λ motivic. ρ λ automorphic representation.. 1. Reductive groups L reductive group. Ψ = (X, Φ, ˇX, ˇΦ). X = Z n, ˇX = Hom(X, Z) X dual, Φ X ˇΦ ˇX. Φ ˇΦ bijection, α Φ bijection ˇα. X ˇX paring,. α Φ s α (x) = x x, ˇα α, x X, 1 F 1 GL(2, F A ) automorphic form. F.. 2

266 256 ˇα ˇΦ sˇα (y) = y y, α ˇα, y ˇX. (1), (2), Ψ root datum. (1) α, ˇα = 2, α Φ. (2) s α (Φ) Φ, sˇα (ˇΦ) ˇΦ, α Φ. Φ Bourbaki root system. F. F F. F 0 F F. G F connected reductive group. G F reductive group. root datum R(G, T ) = (X (T ), Φ, X (T ), ˇΦ). T (F ) G maximal torus, X (T ) = Hom(T, G m ) T character group, Φ root, X (T ) = Hom(G m, T ) T cocharacter group, ˇΦ coroot Ψ root datum Φ reduced. F = F. F connected reductive group G R(G, T ) = Ψ. G F. 1.1 Chevalley. Φ reduced α, nα Φ, 2 n Z. G F Borel B B T. root positive root negative root, simple root. simple root, ˇ simple coroot. R 0 (G, B, T ) = (X (T ),, X (T ), ˇ ), based root datum. R 0 (G) t 1 t 2 G = GL(n), T = t =... t n. ϵ i : t t i T character X (T ) = Zϵ 1 Zϵ 2 Zϵ n 3

267 257. ˇϵ i : u 1... u... 1 ((i, i) u ) T cocharacter. u G m, ϵ i, ˇϵ j = δ ij. X (T ) = Zˇϵ 1 Zˇϵ 2 Zˇϵ n ϵ i (ˇϵ j (u)) = { u, i = j, 1, i j Φ = {ϵ i ϵ j 1 i, j n, i j}, ˇΦ = {ˇϵ i ˇϵ j 1 i, j n, i j}. B positive roots Φ + Φ + = {ϵ i ϵ j 1 i < j n}.. = {ϵ 1 ϵ 2, ϵ 2 ϵ 3,..., ϵ n 1 ϵ n } ˇ = {ˇϵ 1 ˇϵ 2, ˇϵ 2 ˇϵ 3,..., ˇϵ n 1 ˇϵ n }. G Out(G) (1.1) Out(G) = Aut(R 0 (G)) = Aut(G, B, T, {u α } α ). u α 1 α root, Aut(G, B, T, {u α } α ) B, T, {u α } α stabilize Aut(G). (1.1) Inn(G) Aut(G) Out(G) 1 4

268 L-groups F. G F connected reductive group. G F reductive group, F G maximal torus T T Borel B. based root datum R 0 (G) = (X (T ),, X (T ), ˇ ) (2.1) µ G : Gal(F /F ) Aut(R 0 (G)). (2.1). σ Gal(F /F ). σ(b) G Borel, g G(F ) σ(b) = gbg 1. σ(t ) σ(b) maximal torus, σ(t ) = gt g 1. χ X (T ) σχ X (σ(t )) (σχ)(σ(t)) = σ(χ(t)), t T. χ σ (t) = (σχ)(g 1 tg), t T, χ σ X (T ). χ χ σ X (T ), positive roots positive roots. X (T ). 1.1, C connected reductive group L G 0 (2.2) R 0 ( L G 0 ) = (X (G), ˇ, X (T ), ). L G 0 connected L-group. L G 0 G F G = GL(n) L G 0 = GL(n) G semisimple. G simply connected L G 0 adjoint type. G, G L G 0 type A n, D n type A n, D n, type B n, C n. G = SL(n) (type A n 1, simply connected), L G 0 = P SL(n, C) = P GL(n, C) (type A n 1, adjoint type). (1.1) Aut(R 0 ( L G 0 )) = Aut(R 0 (G)) = Out( L G 0 ). (.) (2.1),. 1.3 Gal(F /F ) Out( L G 0 ) µ L G : Gal(F /F ) Aut( L G 0 ) 5

269 259. (µ L G L G 0.) µ L G (2.3) L G = L G 0 Gal(F /F ). G L-group. L-group Weil form variation. Weil. F, K F F Galois. K ab F K Abel. 1 Gal(K ab /K) Gal(K ab /F ) Gal(K/F ) 1, cohomology class η K/F H 2 (Gal(K/F ), Gal(K ab /K)). F non-archimedian local field, [K : F ] = n. H 2 (Gal(K/F ), K ) = Z/nZ, H 2 (Gal(K/F ), K ) fundamental class canonical generator ξ K/F. dense injection K Gal(K ab /K), ξ K/F η K/F. ξ K/F 1 K W F,K Gal(K/F ) 1. W F,K relative Weil. F ur F F. F ur K ab. W F,K Gal(K ab /F ) Gal(F ur /F ) Frobenius. L K F Galois, W F,L W F,K. projective limit Weil W F. W F = lim W F,K. W F Gal(F /F ) Gal(F ur /F ) Frobenius Weil-Deligne group scheme W F. q F, p. g W F F ur Frobenius n, g = q n. R p. W F (R) R W F (x 1, g 1 )(x 2, g 2 ) = (x 1 + g 1 x 2, g 1 g 2 ). W F functor R W F (R) represent Z[1/p] group scheme. (W F, W F (R) inertia group 6

270 260 I, I W F, W F (R).) F archimedean local field. F = C W F = C. F = R W R 1 C W R Gal(C/R) 1. W R Hamilton quaternion algebra H W R = C, j. F global field. C F = F A /F F. [K : F ] = n H 2 (Gal(K/F ), C K ) = Z/nZ, H 2 (Gal(K/F ), C K ) fundamental class canonical generator ξ K/F., F. (Artin map) C K Gal(K ab /K), kernel C K D K. ξ K/F η K/F. ξ K/F 1 C K W F,K Gal(K/F ) 1. W F,K relative Weil. L K F Galois, W F,L W F,K. projective limit Weil W F. W F = lim W F,K. W F Gal(F /F ), ν L G L G = L G 0 W F. L-group Weil form. W F W F,K Gal(K/F ).. (2.3) L-group. Weil form. L-group. canonical homomorphism. π G : L G Gal(F /F ) 2.3. P L G parabolic subgroup P closed subgroup, π G (P ) = Gal(F /F ) P L G 0 L G 0 parabolic subgroup. P( ). P( ) G parabolic subgroup F F 7

271 261. G parabolic subgroup F F P( ) P 0 ( ). ˇ bijection, P( ) P( ˇ ) bijection. bijection P 0 ( ) P 0 ( ˇ ) L G parabolic subgroup P relevant P L G 0 C P 0 ( ˇ ). G F quasi-split, G F Borel subgroup. parabolic subgroup relevant G, H connected reductive group. φ : L H L G L-homomorphism π H = π G φ φ L H 0 : L H 0 L G 0 Lie morphism. 3. Functoriality Principle 1. F local field. F non-archimedean, W F = W F (C), F archimedean W F = W F. G F connected reductive group ϕ : W F L G (i), (ii), (iii), Langlands parameter. (i) W F ϕ L G Gal(F /F ) Gal(F /F ). (ii) ϕ, ϕ(g a ) L G 0 unipotent. ϕ semisimple element semisimple element 2. (iii) ϕ L G parabolic subgroup P, P relevant. L G 0 Langlands parameter. Φ(G) = Φ(G/F ), Langlands parameter. Π(G(F )) G(F ) admissible. Local Langlands Conjecture. ϕ Φ(G) Π ϕ = Π ϕ (G(F )) Π(G(F )) Π(G(F )) = ϕ Φ(G) Π ϕ 2L G L G 0 semisimple, semisimple. F archimedean, W F semisimple. F non-archimedean. W F = C W F (x, g) g = 1, g = 1, x = 0, semisimple. 8

272 262. Π ϕ L-packet. Π ϕ. Π ϕ discrete series Π ϕ discrete series ϕ(w F ) proper Levi subgroup. G = GL(n) Local Langlands conjecture Harris-Taylor- Henniart. Π ϕ. Π ϕ. Arthur [A3]. r : L G GL(n, C) L-group. r L G 0. π Π ϕ L ϵ-factor L(s, π, r) = L(s, r ϕ), ϵ(s, π, r, ψ) = ϵ(s, r ϕ, ψ). ψ F, W F r ϕ L ϵ-factor. G, H F reductive group. L-homomorphism φ : L H L G. ϕ Φ(H) Langlands parameter. φ ϕ : W F L G Langlands parameter 3.1 (i), (ii), G quasi-split (iii). G quasi-split, L-homomorphism functoriality map (3.1) Π ϕ (H) Π φ ϕ (G). G = GL(n) Π φ ϕ (G), Π ϕ (H), (3.1). 2. F global field, G F connected reductive group. v F place. Gal(F v /F v ) Gal(F /F ) 3, inclusion L G v = L (G/F v ) L G. π G(F A ) automorphic representation. π = v π v, π v Π(G(F v )) 3 inclusion v F. 9

273 263 π v. r : L G GL(n, C) L-group. F place v r L (G/F v ) r v, L ϵ-factor L(s, π, r) = v L(s, π v, r v ), ϵ(s, π, r) = v ϵ(s, π v, r v, ψ v ). ψ F A F. L(s, π, r) Re(s). L(s, π, r) s L(s, π, r) = ϵ(s, π, r)l(1 s, π, r). ( contragredient.) H F connected reductive group L-homomorphism φ : L H L G. F place v L-homomorphism φ v : L (H/F v ) L (G/F v ). ρ = v ρ v H(F A ) automorphic representation. Local Langlands conjecture. Langlands parameter ϕ v Φ(H/F v ) ρ v Π ϕv. G F quasi-split. Problem on Functoriality. G(F A ) automorphic representation π = v π v π v Π φv ϕ v, v., functoriality correspondence A(H A ) ρ π A(G A ) local ρ v π v (3.1) consistent. π ρ functorial image. ( local L-packet, π.) (conjectual answer) π m(π).. (.) π global Langlands parameter ϕ. ϕ tempered parameter. Labesse-Langlands-Kottowitz m(π) = S ϕ 1 x S ϕ ϵ ϕ (x) x, π, x, π = v x, π v v 10

274 264. local L-packet generic π generic. x, π = 1 x S ϕ ϵ ϕ (x) 0. G = GL(n) S ϕ = Examples. (1) D F quaternion algebra. D D F H. H F -algebra A H(A) = (D F A). L H = GL(2, C) Gal(F /F ). G = GL(2)/F. L H = L G L- homomorphism correspondence A(H A ) v π v v π v A(G A ). D v split, D(F v ) = GL(2, F v ) π v = π v. Jacquet-Langlands-Shimizu correspondence. π = v π v image (π ), D v division algebra v π v principal series. (2) [F : Q] = 2, H = GL(2)/Q, G = Res F/Q (GL(2)). L-group finite form 4 L G = GL(2, C) 2 Gal(F/Q), L H = GL(2, C) Gal(F/Q). σ Gal(F/Q). L-homomorphism L H L G L H (g, σ i ) (g, g, σ i ) L G. L G r : L G GL(4, C) ( ) r((g 1, g 2 ), σ i g1 0 ) = w i, w = 0 g 2 ( 0 ) ρ A(H A ) functorial image π A(G A ) L(s, π, r) = L(s, ρ)l(s, ρ χ) 4 Restriction of scalars L-group induced group. [B]. 11

275 265. χ F/Q F A. G A = G(Q A ) = GL(2, F A ) π GL(2, F A ) ρ L(s, π, r) = L(s, ρ) L(s, ρ) = L(s, ρ)l(s, ρ χ). ρ ρ lifting. Gal(F/Q). (3) H = GL(2)/F, G = GL(n + 1)/F. L-homomorphism L H L G L H (g, σ) (ρ n (g), σ) L G. ρ n n. functorial image n = 2 Gelbart-Jacquet, n = 3 Shahidi-H.Kim, n = 4 H. Kim. functoriality n Ramanujan, Selberg, Sato-Tate,. n 3 endoscopic lift.. (4) G = GSp(n). L G 0 = (GL(1, C) Spin(2n + 1, C))/A, A = {1, a}. a = (a 1, a 2 ), a 1 = 1 GL(1, C), a 2 Spin(2n + 1, C) center 2. π G A automorphic representation. Spin(2n + 1, C) (2n + 1) (standard representation) st L. (st Spin(2n + 1, C) SO(2n + 1, C).) L(s, π, st) π standard L. Spin(2n + 1, C) spinor spin. 2 n. L(s, π, χ spin) π spin L. χ GL(1, C) 2 χ( 1) = 1. n 4 L(s, π, χ spin). 4. E F. M F motive, E. E finite place λ λ-adic representation ρ λ : Gal(F /F ) GL(d, E λ ). ρ λ automorphic representation. [Y]. M Betti realization H B (M) E d. d M rank. H B (M) Q C Hodge. M weight, M pure weight. w. Hodge, Hodge Hg(M). Hg(M) E connected group. M polarizable. Hg(M) reductive. H Im(ρ λ ) Zariski closure. 12

276 H E λ. H 0 H, H 0 = Hg(M). E = Q. E = Q, compatible.. F Galois K H 0 ρ λ (Gal(F /F )) = ρ λ (Gal(F /K)). E λ C, (4.1) 1 H 0 (C) H(C) Gal(K/F ) 1. (4.1) H 0. φ : Gal(K/F ) Out(H 0 ) = Aut(R 0 (H 0 )) 4.2. F connected reductive group G. (i) G F quasi-split. (ii) L G 0 = H 0 (C). (iii) µ G = φ. M automorphic representation G A. 5 σ Gal(K/F ), σ H(C) σ (4.1) σ. { f(σ, τ) στ = σ τ, a(σ)n = σn σ 1, n H 0 (C). a(σ) Aut(H 0 (C)). {a(σ), f(σ, τ)} { i(f(σ, τ))a(στ) = a(σ)a(τ), (4.2) f(σ, τ)f(στ, ρ) = (a(σ)f(τ, ρ))f(σ, τρ). i(f(σ, τ)) f(σ, τ) Inn(H 0 ) Aut(H 0 ) Out(H 0 ) 1 split, π : Aut(H 0 ) Out(H 0 ) canonical homomorphism, s : Out(H 0 ) Aut(H 0 ) π s = id. α σ H 0 (C) (4.3) s(π(a(σ))) = i(α σ )a(σ) 5 G = GL(d)/E. minimal. minimal GL(d). 13

277 267. σ α σ σ, {a(σ), f(σ, τ)} {a Z (σ), f Z (σ, τ)}. { a Z (σ) = i(α σ )a(σ), f Z (σ, τ) = α σ (a(σ)α τ )f(σ, τ)α 1 στ. (4.3) σ a Z (σ). (4.2) i(f Z (σ, τ)) = 1. f Z (σ, τ) H 0 (C) Z(H 0 (C)). 2-cocycle f Z Z 2 (Gal(K/F ), Z(H 0 (C))) f Z cohomology ( H 2 (Gal(K/F ), Z(H 0 (C)))) M. ρ λ f Z cohomology, σ, α σ, s. 4.1 λ. ρ λ Gal(F /F ) abelian, fundamental class 6, cohomology. v F finite place. ρ λ λ-adic representation ρ λ,v : Gal(F v /F v ) H(E λ ) GL(d, E λ ). ψ v : W F v H(E λ ) H(C). f Z L. L K F Galois, f Z Gal(L/F ) inflate 2-cocycle split. 1 L G 0 L G Gal(L/F ) 1 1 H 0 (C) H(C) Gal(K/F ) 1. ψ v Langlands parameter ϕ v : W F v L G. v infinite place, H B (M) Hodge Langlands parameter ϕ v automorphic representation π = v π v A(G A ) π v Π ϕv, v. f Z splitting field. [Y].. 6 Weil. 14

278 Langlands functoriality (automorphic representation ) functorial.. References [A1] J. Arthur, Unipotent automorphic representations: Conjectures, Astérisque, (1989), [A2] J. Arthur, A note on the automorphic Langlands group, Canad. Math. Bull. 45 (2002), [A3] J. Arthur, A note on L-packets, Pure and applied Mathematics Quarterly 2 (2006), [B] A. Borel, Automorphic L-functions, Proc. Sympos. Pure Math. 33 (1979), part 2, [V] D. Vogan, The local Langlands conjecture, Contemp. Math. 145 (1993), [Y] H. Yoshida, Motivic Galois groups and L-groups, Clay Mathematics Proceedings, 13 (2011), Summer School resume.. Corvallis conference Proc. Sympos. Pure Math. 33 (1979).. The map of my life, Springer, 2008,,, [64e]. S. Lang, Some history of the Shimura-Taniyama conjecture, Notices of AMS. 42 (1995), ,.., Borel, Humphreys. Springer.. reductive group Borel-Tits A. Borel and J. Tits, Groupes réductifs, Publ. Math. IHES 27 (1965),

279 269. SGA3 scheme group scheme,. reductive group Chevalley. Root N. Bourbaki, Groupes et algèbres de Lie, Chapitre IV, V, VI.. Weil Artin-Tate Class field theory. Weil. Weil C K D K Riemann,. D K = {1} Gal(K ab /F ) Weil,. D K. L-group References Vogan. L-group Local Langlands Conjecture.. multiplicity formula [A1] J. -P. Labesse and R. P. Langlands, L-indistingushability for SL(2), Can. J. Math. 31 (1979), , R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), ,. Arthur. Net Arthur Arxiv... Motive AMS, Proc. Symposia pure Math. Vol. 55 (Part 1) Motive.. Hodge group Springer Lecture Note 900 Deligne (Milne ). Motive category U. Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107 (1992), Motive mixed motive scheme motive. algebraic cycles (Hodge, Tate ). 16

280 270 Seesaw dual pair., seesaw dual pair,,. Weil Howe,,., seesaw dual pair Siegel-Weil, L,.,, Rallis, L, L Jacquet, L. seesaw dual pair, Kudla [19]..,. 1 Seesaw dual pair seesaw 1.1 Seesaw dual pair F., F 0. G = Sp N (G, H), G G H, H G G, dual pair G dual pair (G, H), (G, H ) G G H H, seesaw dual pair (1) 1.1, dual pair (weakly dual pair)., Kudla [19]. (2), : H H G G

281 Seesaw dual pair 1 ( )., : W W F F W, (, ) : V V F F V, O(V )., = (, ),, W = V F W, (Sp(W ), O(V )) Sp(W) dual pair. W W = W 1 +W 2, seesaw pair : Sp(W ) O(V ) O(V ) Sp(W 1 ) Sp(W 2 ) O(V ), V = V 1 + V 2 V, seesaw pair : Sp(W ) Sp(W ) O(V ) Sp(W ) O(V 1 ) O(V 2 ) W = W W r, V = V V s. seesaw pair. 2. K/F, σ Galois Gal(E/F ). (W,, ) K, (V, (, ) ) K, U(W ) U(V )., = Tr K/F ((, ), σ ), W = V K W, (U(W ), U(V )) Sp(W ) dual pair. F (V, (, )), V σ-,, V V F K, (x α, y β) = α σ (x, y)β, (x, y V, α, β K). R = R K/F F, F (R K/F W, Tr K/F, )., W V F R K/F W, seesaw pair : U(V ) Sp(R K/F W ) O(V ) U(W ) 3. U 1 U 2 F, U 1 U 2. U 1 U 2 U 1 U 2 F W = U 1 U 2 + U 1 U 2, (GL(U 1 ), GL(U 2 )) Sp( W ) dual pair. 1

282 272 (W,, ) polarization, V V, W (W ) (, ),,, W V W + V W V (W ) + V W, seesaw pair : GL(V ) Sp(W ) O(V ) GL(W ) 1.3 F, F ψ. G G, G G G = Mp N G. (G, H) G dual pair, Weil ω ψ G H. G (genuine) π, ωψ [π] = ω ψ / f ker f., f 0 G ω ψ π. G H, H (genuine) Θ ψ (π), ω ψ [π] π Θ ψ (π)., ψ. 1.3 ( Howe ). Θ(π), 0, θ ψ (π). π θ ψ (π) (1) Howe. 2 [37]. F [12]., Howe. (2) (G, H) dual pair, Mp(W ) Sp(W ) G H ( [25, 20] ). Mp(W ),.. Kudla [20]., G H Mp(W )., G ( H) genuine G ( H), Howe G H. 1.5 ( seesaw ). (G, H), (G, H ) seesaw dual pair, π G, σ H, : Hom G (Θ(σ), π) Hom H (Θ(π), σ).

283 273 Proof. Hom G H (ω, π σ) Hom G H (π Θ(π), π σ) Hom H (Θ(π), σ).,. Hom G H (ω, π σ) Hom G (Θ(σ), π) 1.4 F, A, ψ A/F. GA G = Sp(W) G(A), (ω ψ, S) G A Weil. (G, H) G dual pair. (π, V π ) G(A). π v θ ψv (π v ), v θ ψv (π v ). 1.6 ( Howe ). v θ ψv (π v ) H(A).,. W polarization W = X + Y, Schwartz S(X(F v )) Weil, Weil Schrödinger. K v G(F v ), K v H(F v). v, g v G(F v ), h v H(F v ), Schwartz S(X(F v )) K v K v S(X(F v)) (g v, K v ) (h v, K v)- ( Fock ). v, S(X(F v )) = S(X(F v )), S(X(A)) = v S(X(F v )). G A G(A) G(F ) G A, Schwartz ϕ S(X(A)), Θ(ϕ)(g) = x X(F ) (ω ψ (g)ϕ)(x) g G A γ G(F ), Θ(ϕ)(γg) = Θ(ϕ)(g) ( ). ξ V π θ ϕ (ξ)(h) = ξ(g)θ(ϕ)(g, h)dg G(F )\G(A) H(A). H(A) θ ψ (π) = {θ ϕ (ξ) ϕ S(X(A)), ξ V π } π H. θ ψ (π), 0, v θ ψv (π v ) ( ), Howe., θ ψ (π).

284 274 ξ 1, ξ 2 G(A),, ξ 1, ξ 2 G. ξ 1, ξ 2 G = ξ 1 (g)ξ 2 (g)dg. G(F )\G(A). 1.7 ( seesaw ). (G, H), (G, H ) seesaw dual pair, (π, V π ) G(A), (σ, V σ ) H (A), (ξ, ξ ) V π V σ ϕ S(X(A)), : θ ϕ (ξ), ξ H = θ ϕ (ξ ), ξ G (1) θ ϕ (ξ) θ ϕ (ξ ) H (A) G(A). Θ(ϕ)(g, h) Θ(ϕ)(g, h ) Θ(ϕ)(g, h ), Θ(ϕ)(g, h)., 1.7. (2), 1.7 ξ ξ.,. 2 seesaw machine 2.1 Siegel-Weil Siegel-Weil, Eisenstein,, Siegel [36], Weil [39]., Kudla Rallis[22], regularized Siegel-Weil. seesaw, regularized Siegel-Weil,., [22] [40, 41]. G = Sp 2n, H = O(V ), m = dim V. F. (G, H) G dual pair. G H Mp(W ) G Weil ω, G H. X = V n, Schrödinger S(V n ) O(V ), ω(h)ϕ(x) = ϕ(h 1 x). V F χ V. G P ( ( ) ) ω a 0 0 t a 1 Φ(x) = χ V (det a) det a m/2 Φ(xa), a GL n (F ), ( ( ) ) ω 1 b Φ(x) = ψ(tr(b(x, x))/2)φ(x), b Sym 0 1 n (F )., x = (x 1,..., x n ) V n, (x, x) = 1 2 ((x i, x j )) Sym n (F ). I(s) = Ind G P ((χ V s ) det) { = f (s) : G C f (s)( ( ) ) } a b 0 t a 1 g = χ V (det a) det a s+(n+1)/2 f (s) (g),

285 275 s 0 = (m n 1)/2, G-, H- S(V n ) I(s 0 ), Φ f (s 0) (g) = (ω(g)φ)(0) Φ. Rallis [30, 25],, H G Θ ψ (1l). G = O(n, n), H = Sp 2j, I(s)., F, V F. Φ S(V n (A)), G(A) I(s) f (s) Φ f (s 0) Φ = v f (s 0) Φ v. I(s) f (s), E(f (s) )(g) = f (s) (γg) γ P (F )\G(F ). Langlands (cf. [2]), E(f (s) ), E(f (s) ) = E(M(s)f (s) )., M(s) : I(s) I( s), N P, w G(F ) Weyl, M(s)f (s) (g) = f (s) (wug)du N(F )\N(A)., θ ϕ (1)(g) = H(F )\H(A) Θ(ϕ)(g, h)dh H 1l. 1l, ( ). 2.1 (Siegel-Weil )., ϕ S(V n (A)) E(f (s) ϕ ) s=s 0 = θ ϕ (1) (1), s 0. Weil [39, Proposition 8], regularize. (2), [14, 40, 41] (1) F = Q, (, ), L V (F ) (, m 8 ). {L j } L, ϵ(l j ) = #O(V ) GL(L j ), θ Lj (Z) = e π 1tr((x,x)Z), Z Sym n (C), IZ > 0 x L n j, E m/2 (Z) = j ϵ(l j) 1 θ Lj (Z) j ϵ(l j) 1

286 ( : ϕ = e πtr((x,x)), ϕ L Ẑ, ϕ = ϕ ϕ, 2.1 ). (2) (1) Fourier ( [34, 35] ), Siegel : H H, H(A) f H P H (f) = f(h)dh H (F )\H (A). f 0, P H (f) 0. H(A) (π, V π ), f V π P H (f) 0, H, P H (π) 0. H H, H L., seesaw pair.,. (π, V π ) G(A), (π, V π ) π. V π = V π. seesaw. θ G θ ϕ(1) = E(f (s) ϕ (ξ) H ϕ ) s=s 0 H 1l G ξ Vπ seesaw, Siegel-Weil P H (θ ϕ (ξ)) = θ ϕ (ξ), 1 H = ξ, E(f (s) ϕ ) G s=s0 = lim s s0 G(F )\G(A) ξ(g)e(f (s) ϕ )(g)dg. π, Rankin-Selberg Z(ξ, f (s) ) = ξ (g)e(f (s) )(g)dg (ξ V π, f (s) I(s)) G(F )\G(A) : Eisenstein, Z(ξ, f (s) ) E(f (s) ),,.. Z(ξ, f (s) ) = Z(ξ, M(s)f (s) )

287 277 (1) ξ = v ξ v f (s) = v f v (s), Euler : Z(ξ, f (s) ) = v Z v (ξ v, f (s) v ). (2) ξ 0,v, f (s) 0,v, L r :L G GL N (C), π v Langlands L : Z v (ξ 0,v, f (s) 0,v ) = L(s + 1/2, π v, r). (3) F v, G(F v ) π v,, Γ(s, π v, ψ v ), Z v (ξ v, f (s) ) = Γ ( s + 1 2, π v, ψ v ) Zv (ξ v, M v (s)f (s) ) (ξ v π v, f (s) v I v (s))., M v (s) : I v (s) I v ( s) M(s). Euler (2),., S, Z(ξ, f (s) ) = L S( s + 1 2, π, r) v Z v(ξ v, f (s) v ) (Z(ξ, f (s) ) s s, L S (s, π, r) s 1 s ). L,., L, L, L : L, ε, γ F v, subscript v., F = F v, G = G(F v ), I(s) = I v (s), π G. M(s) M (s), γ., Z(ξ, M (s)f (s) ) = γ ( s + 1 2, π, ψ) Z(ξ, f (s) ). γ (, )., π Ind G P σ γ(s, π, ψ) = γ(s, σ, ψ) (Shahidi [33] Lapid Rallis [24] Ten commandments )., L ε. K- f (s) : G C C s s C f (s) I(s), I(s). I(s) f (s) f (s) 1 f (s) 2, f (s) = f (s) 1 + M ( s)f ( s) 2,. :

288 ([11, 42]).. f (s). f (s) Rs 0 M (s)f (s) Rs < 0. Tate L Euler, : (1) π, Euler L(s, π) ε(s, π, ψ),. ξ π f (s), Z(ξ, f (s) )/L(s + 1/2, π). s C, ξ π f (s), Z(ξ, f (s) )/L(s + 1/2, π) s=s 0. ξ π f (s) : Z(ξ, M (s)f (s) ) L ( 1 2 s, π ) = ε( s + 1 2, π, ψ) Z(ξ, f (s) ) L ( s + 1 2, π). (2) π, L(s, π) = L(s, π, r). ψ, ε(s, π) = 1. (3) : 2.5. Z (ξ, f (s) ) = Z(ξ, f (s) )/L(s + 1/2, π). s, Z (s ) : (ξ, f (s ) ) Z (ξ, f (s ) ) 0 Hom G (π I(s ), C) Hom G (I(s ), π) 0. (1) : L(s, π) ε(s, π, ψ) = γ(s, π, ψ) L(1 s, π ). (2) 3(2) 2(2). (3) L., L. (4) s, dim Hom G (π I(s ), C) 1, 2(3)., [27] Part B, 12 Bernstein. F, π G(A), L ε L(s, π) = v L(s, π v ), ε(s, π) = v ε(s, π v, ψ v ). s, L(s, π). ε(s, π), ψ., L(s, π), L(s, π) = ε(s, π)l(1 s, π), L(s, π) E(f (s) ).

289 :, s 0 = 0., ξ = v ξv ϕ = v ϕ v S(X(A)), π P H (θ ϕ (ξ)) = L(1/2, π) v Z v (ξ v, f (0) ϕ v ). θ(π) H, : ( ) L(1/2, π) 0; ( ) v, Z v (0) Θv (1l) 0.,. seesaw Z v (0) Θv (1l) Hom G (Θ v (1l), π v ) Hom H (Θ v (π v ), C),, Hom H (Θ v (π v ), C) 0., : H + H + G G H G H G (H +, H + ) (H, H ). ϵ = ±, H ϵ(f v ) π G (F v ) Θ ϵ v(π),. I v (0) = Θ + v (1l) Θ v (1l) (2.1) dim Hom G (I v (0), π) = dim Hom G (Θ + v (1l), π) + dim Hom G (Θ v (1l), π) = dim Hom H + (Θ + v (π), C) + dim Hom H (Θ v (π), C) Hom H + (Θ + v (π), C) Hom H (Θ v (π), C) 0,. 2.6 Gross-Prasad, : 2.6. π G(A). (1) : P H (θ(π)) 0;

290 280 L(1/2, π) 0 v Hom G (Θ(π v ), C) 0. (2), : H 0 H 0, π H 0 H 0 ; L(1/2, π) 0. Proof. Z v (0) Θ ϵ v (1l) Hom H ϵ (Θ ϵ v(π), Z v (0) Θ ϵ v (1l) 0 Hom H ϵ (Θ ϵ v(π), C) 0., (2.1) Zv (0) Θ + v (1l) 0 Z v (0) Θ v (1l) 0,., (1). (2.1) I(0) = R + R, R ϵ = (ϵv) v: Q v ϵv=ϵ Θ ϵv v (1l).. M(s) s = 0 R 1 (cf. 2.7(2)), R Eisenstein,. H F H i G Θ H i (1l), Minkowski-Hasse ( ), R + = H i Θ H i (1l). L L(1/2, π) = i Z(ξ i, f (s) i ) s=0 = i Z(ξ i, f (s) ϕ i ) s=0 = P H i (θ ϕi (ξ i )). i P H i (θ ϕi (ξ i )) (1) θ(π) H, Hom H (A)(Θ(π), C) 0. F v Hom H (F v)(θ v (π v ), C) 0. (2) M v (s) s = 0, Θ ϵ v(1l) 1, : Hom H ϵ (Θ ϵ v(π), C) 0 ε(1/2, π, ψ v ) = M (0) Θ ϵ v (1l)., Hom H ϵ (Θ ϵ v(π), C) 0 ϵ ε(1/2, π, ψ v ).. (3) L, R Eisenstein. Eisensetein,. [23].

291 281 3 seesaw machine 3.1 1: doubling seesaw Rallis doubling,. F, (V, (, )) n, V = V V, V 1 = V {0}, V 2 = {0} V. (, ) : V V F : (x + y, x + y ) = (x, x ) (y, y ) (x, x V 1, y, y V 2 ). G = O(V ) G = O(V ) V V. G(A) π Sp 2j θ j (π). : 3.1 (Rallis [30]). (1) j n, θ j (π) 0. (2) θ j (π) 0, j j, θ j (π) 0. (3) j 0 θ j0 (π), θ j0 (π)., j = (n 1)/2, H = Sp n 1, H = Mpn 1, θ(π) = θ n 1 (π), seesaw. Sp(W ) Sp(W ) G Sp(W ) G G Weil ω ψ G H ω ψ,v ( (, ) ). a F, ψ a (x) = ψ(ax), av = (V, a(, )). c a GSp 2n(n 1) similitude a, ω ψa,v ω ψ,av ω c a ψ,v. seesaw machine (s) θ G E(f σ(ϕ 1 ϕ ) ϕ1 (ξ 1 )θ ϕ2 (ξ 2 ) Sp(W ) Sp(W ) 2 ) s=0 1l Sp(W ) G G ξ1 V π, ξ 2 V π (σ Fourier ), θ ϕ (ξ), θ ϕ (ξ) H, θ ϕ (ξ), θ ϕ (ξ) H = Z(ξ ξ, f (s) σ(ϕ ϕ) ) s=0.

292 282, ξ V π, ξ V π, I(s) f (s), : Z(ξ ξ, f (s) ) = ξ(g)ξ (g )E((g, g ); f (s) )dg 1 dg 2. G(F ) G(F )\G(A) G(A), Eisenstein., Kudla Rallis [22], Wee Teck Gan [6] [40, 41]., Wee Teck Gan (, ): V L, n, Sp(n 1, C) Gal(F /F ), n, SO(n, C) Gal(F /F ). n V E F, ϵ = diag[1, 1,..., 1, 1] O(n, C) \ SO(n, C), Gal(E/F ), g ϵgϵ 1.. n, N = n 1, n, N = n, std : L G GL N (C), L (1) G L. Adams [1], G L n, Sp(n 1, C) Z/2Z Gal(F /F ), n, O(n, C) Gal(F /F ). (2) (, ) = 0, G = GL(V ), L Godement-Jacquet L ( [7] ). P = {g G V g = V } G. coset G (F ) = i P (F )γ i(g(f ) G(F )), ξ(g)ξ (g ) f (s) (γ(g, g ))dgdg G(F ) G(F )\G(A) G(A) γ P (F )\P (F )γ i(g(f ) G(F )). main orbit P (F )\G(F ) G(F ) = G(F ) e 0, Z(ξ ξ, f (s) ) = π(g)ξ, ξ G f (s) ((g, e))dg G(A) (, [27] ). π v π v, π π v, I(s) v I v (s). ξ v π v, ξv πv, f v (s) I v (s), Z(ξ v ξv, f v (s) ) = π v (g)ξ v, ξv f v (s) ((g, e))dg G(F v). π v ξ v,0, ξ v,0, f (s) v,0, Z(ξ v,0 ξ v,0, f (s) v,0 ) = L(s, π v, std) ξ v,0, ξ v,0 b v (s) 1, b v (s) = [n/2] j=1 ζ v (2s + n + 1 2j).

293 283., ξ = v ξ v, ξ = v ξv, f (s) = v f v (s), S, Z(ξ ξ, f (s) ) = v S Z(ξ v ξv, f v (s) ) L S (s, π, std)/b S (s). γ [24]. L ε [26, 11, 42]., L L(s, π) (,, ) Eisenstein. 3.3 ([21, 42]). L(s, π) Rs > n 2,. L(s, π) X = { 1 n 2, 2 n 2,..., } { n 2 \ 1 } 2, L(s, π) = ε(s, π)l(1 s, π) GL N L, L.. [22, 6, 42]. seesaw, 0 Hom Mpn 1 (Θ(π v ) Θ(π v ), C) 0, π v 0. v, sgn v : O(V v ) µ 2 determinant. T, sgn T = v T sgn v. T, sgn T. v, L(s, π v sgn v ) = L(s, π v ), ε(s, π v sgn v, ψ v ) = ε(s, π v, ψ v ), #T L(s, π sgn T ) = L(s, π).,. 3.5 (Gan-Savin [5]). F, π G. Θ + (1l) = Θ(1l), Θ (1l) = Θ(1l) sgn, Θ + (π) = Θ(π), Θ (π) = Θ(π sgn)., Θ + (π) Θ (π sgn) 0. Z (0) Θ ϵ (1l) 0 Θ ϵ (π) ,. 3.6 (cf. [41]). π O(V ), L(s, π)., : θ(π) 0;

294 284 L(1/2, π) 0 v, θ v (π v ) 0; L(1/2, π) 0 v, π v ( 1) = ε(v v )ε(1/2, π v ). L(1/2, π) 0, T, θ(π sgn T ) L, θ(π), Rallis : L B F, Z F B, (π, V π ) B (A). π χ π, χ π = 1. E F, E B., B b, B = E Eb, T = E B. Waldspurger [38], f V π P T (f) = f(t)dt Z(A)T (F )\T (A). O(B) B, seesaw similitude., O(B) SL 2 SL 2 O(E) O(Eb) SL 2 GSO(B) (GL 2 GL 2 ) 0 (GSO(E) GSO(Eb)) 0 GL 2 (GL 2 GL 2 ) 0 = {(g, g ) GL 2 GL 2 det g = det g }. (GSO(E) GSO(Eb)) 0. π GL 2 (A), χ π = 1. GL 2 (A) π GL 2 (A) I(s), : Z(f, f ; f (s) ) = Z(A)GL 2 (F )\GL 2 (A) f(g)f (g)e(f (s) )(g)dg (f V π, f V π, f (s) I(s)).

295 285 Jacquet [18], M 2 (C) M 2 (C) M 4 (C) r : GL 2 (C) GL 2 (C) GL 4 (C) GL 2 GL 2 L L(s, π π, r). accidental GSO(E) E, GSO(B) B B /{(z, z 1 ) z F }.. Jacquet-Langlands, π B π B (π B, π B = 0 ),, θ(π) π B (π B ) π B π B. seesaw machine, ξ V π, ξ B 1, ξ B 2 V π B, P T (ξ1 B )P T (ξ2 B ) = Z( ξ, E(f (s) ϕ ) s=0; f (s) ϕ ) s=0 = v Z v ( ξ v, f (0) ϕ v ; f (s) ϕ ) s=0 v = L S (1/2, π θ(1l), r) Z v ( ξ v, f (0) ϕ v ; f (s) ϕ v v S = L(1/2, BC E (π)) Zv ( ξ v, f (0) ϕ v ; f (s) ϕ v v S ) s=0 ) s=0. BC E (π) π GL 2 (E). (GL 2 GL 2 ) 0 E(f (s) (s) ϕ )E(f GSO(B) 1l (GSO(E) GSO(Eb)) 0 GL 2 ξ V π ϕ ) s=0 Weil Siegel-Weil similitude, [3] (Tunnel, H. Saito [32]). F, GL 2 (F ) π Jacquet-Langlands π JL ( π JL = 0 ) dim Hom E (π, C) + dim Hom E (π JL, C) = 1. Waldspurger : 3.9 (Waldspurger [38]). π GL 2 (A), E F. : L(1/2, BC E (π)) 0; B E, f V π B P T (f) 0.

296 286 E B, : P T (π B ) 0; L(1/2, BC E (π)) 0 v Hom E v (π B v v, C) 0., P T (f) 2. [13] Wee Teck Gan [3], : L Jacquet (π i, V πi ) (i = 1, 2, 3) GL 2 (A), χ π1 χ π2 χ π3 = 1. B F, πi B 3.2. fi B πi B, I(f1 B, f2 B, f3 B ) = f1 B (g)f2 B (g)f3 B (g)dg Z(A)B (F )\B (A). r : GL 2 (C) GL 2 (C) GL 2 (C) GL 8 (C). Jacquet, Harris Kudla [9, 10] (Jacquet Harris-Kudla )., : L(1/2, π 1 π 2 π 3, r) 0; B, f B i V π B i I(f B 1, f B 2, f B 3 ) 0., B, : v Hom B v (π B v 1,v πb v 2,v πb v 3,v, C) (1) 3.6, ,. (2) , Gross-Prasad [8]. Gross-Prasad, [13]. L L(s, π 1 π 2 π 3, r), Garrett [4], Piatetski-Shapiro Rallis [28] [15, 16, 17], Ramakrishnan [31], L ε. Z(f 1, f 2, f 3 ; f (s) ) = f 1 (g 1 )f 2 (g 2 )f 3 (g 3 ) Z(A)(GL 2 (F ) 3 ) 0 \(GL 2 (A) 3 ) 0 E(f (s) )(g 1, g 2, g 3 )dg 1 dg 2 dg 3 (f i V πi, f (s) I(s)) , L L(s, π 1 π 2 π 3, r). seesaw machine

297 287 f1 B, f2 B, f B 3 (GSO(B) 3 ) 0 GSp E(f (s) 6 ϕ ) s=0 GSO(B) (GL 1l 3 2) 0 f 1, f 2, f 3, f i V πi, f B i V π B i, I(f B 1, f B 2, f B 3 ) 2 = Z( f 1, f 2, f 3 ; f (s) ϕ ) s=0 = v Z v ( f 1,v, f 2,v, f 3,v ; f (s) ϕ v ) s=0 = L(1/2, π 1 π 2 π 3, r) v S Z v ( f 1,v, f 2,v, f 3,v ; f (0) ϕ v ) s=0. (π 1 π 2 π 3 ) π 1 π 2 π 3. Weil Siegel-Weil similitude, [9]. Prasad [29] (D. Prasad [29]). F, B F, GL 2 (F ) (π 1, π 2, π 3 ) dim Hom GL2(F )(π 1 π 2 π 3, C) + dim Hom B (π B 1 π B 2 π B 3, C) = 1., References [1] J. Adams, L-functoriality for dual pairs, Astérisque (1989) [2] J. Arthur, Eisenstein series and the trace formula, Proc. Symp. Pure Math. 33 (1979) part 1, [3] W. T. Gan, The Shimura correspondence a lá Waldspurger, preprint. [4] P. Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. of Math. 125 (1987) [5] W. T. Gan and G. Savin, Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence, preprint. [6] W. T. Gan and S. Takeda, The regularized Siegel-Weil formula: the second term identity and non-vanishing of theta lifts from orthogonal groups, J. Reine Angew. Math. 659 (2011) [7] R. Godement and H. Jacquet, Zeta functions of simple algebras, Springer Lec. notes in Math., vol. 260, Springer-Verlag, Berlin, [8] B. H. Gross and D. Prasad, On the decomposition of a representation of SO n when restricted to SO n 1, Canad. J. Math. 44 (1992) [9] M. Harris and S. Kudla, The central critical value of a triple product L-function, Ann. of Math. (2) 133 (1991)

298 288 [10] M. Harris and S. Kudla, On a conjecture of Jacquet, in Contributions to Automorphic Forms, Geometry, and Number Theory (Baltimore, 2002), Johns Hopkins Univ. Press, Baltimore, 2004, [11] M. Harris, S. Kudla and W. J. Sweet Jr., Theta dichotomy for unitary groups, J. Am. Math. Soc., 9 (1996) [12] R. Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989) [13] A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, Geom. Funct. Anal. 19 (2010) [14] A. Ichino, A regularized Siegel-Weil formula for unitary groups, Math. Z. 247 (2004) [15] T. Ikeda, On the functional equations of triple L-functions, J. Math. Kyoto Univ. 29 (1989) [16] T. Ikeda, On the location of poles of the triple L-functions, Compos. Math. 83 (1992) [17] T. Ikeda, On the gamma factor of the triple L-function I, Duke Math. J. 97 (1999) [18] H. Jacquet, Automorphic forms on GL(2) Part II, Springer Lec. notes in Math. 278, [19] S. Kudla, Seesaw dual reductive pairs, Automorphic forms of several variables. Prog. Math. vol. 46, pp Boston, MA: Birkhäuser [20] S. Kudla, Splitting metaplectic covers of dual reductive pairs, Isr. J. Math. 84 (1994) [21] S. Kudla and S. Rallis, Poles of Eisenstein series and L-functions, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II, , Israel Math. Conf. Proc. 3, Weizmann, Jerusalem, [22] S. Kudla and S. Rallis, A regularized Siegel-Weil formula: the first term identity, Ann. Math. 140 (1994) [23] S. Kudla, M. Rapoport and T. Yang, Modular forms and special cycles on Shimura curves, Annals of Math. Studies 161, Princeton Univ. Press, [24] E. Lapid and S. Rallis, On the local factors of representations of classical groups, Automorphic representations, L-functions and applications: progress and prospects, Berlin: de Gruyter (2005) [25] C. Moeglin, M.-F. Vignera and C.-L. Waldspurger, Correspondence de Howe sur un corps p-adique, Springer Lec. notes in Math. 1291, [26] I. Piatetski-Shapiro and S. Rallis, ε factor of representations of classical groups, Proc. Nat. Acad. Sci. U.S.A., 83 (1986) [27] I. Piatetski-Shapiro and S. Rallis, L-functions for classical groups, in Springer Lec. notes in Math., vol (1987) [28] I. Piatetski-Shapiro and S. Rallis, Rankin triple L-functions, Compos. Math. 64 (1987)

299 289 [29] D. Prasad, Trilinear forms for representations of GL(2) and local epsilon factors, Compos. Math. 75 (1990) [30] S. Rallis, On the Howe duality conjecture, Compos. Math. 51 (1984) [31] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. (2) 152 (2000) [32] H. Saito, On Tunnell s formula for characters of GL(2), Compos. Math. 85 (1993) [33] F. Shahidi, A proof of Langlands conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. (2) 132 (1990) [34] G. Shimura, On Eisenstein series, Duke Math. J. 50 (1983) [35] G. Shimura, The number of representations of an integer by a quadratic form, Duke Math. J. 100 (1999) [36] C. L. Siegel, Indefinite quadratische Formen und Funktionentheorie I, Math. Ann. 124 (1951) 17 54; II, (1952) [37] J.-L. Waldspurger, Démonstration d une conjecture de dualité de Howe dans le cas p-adique, p 2, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Pt. I: Papers in representation theory, Pap. Workshop L- Functions, Number Theory, Harmonic Anal., Tel-Aviv/Isr. 1989, Isr. Math. Conf. Proc. 2, , Weizmann, Jerusalem [38] J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math. 54 (1985) [39] A. Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965) [40] S. Yamana, On the Siegel-Weil formula: The case of singular forms, Compos. Math. 147 (2011) [41] S. Yamana, On the Siegel-Weil formula for quaternionic unitary groups, Am. J. Math. (to appear) [42] S. Yamana, L-functions and theta correspondence for classical groups, (preprint)

300 290 Theta Cohomological ( ) 1, (derived functor module), (A q (λ)) Knapp-Vogan [KV] Anthony W. Knapp and David A. Vogan, Jr. Cohomological induction and unitary representations, volume 45 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, ( 2, 3).. A q (λ), [VZ] David A. Vogan, Jr. and Gregg J. Zuckerman. Unitary representations with nonzero cohomology. Compositio Math., 53(1):51 90, 1984., A q (λ).,. [KV], (g, K)- C(g, K),.,, θ-stable (q, L K) A q (λ). Appendix.,. 4.. [Kn] Anthony W. Knapp. Representation theory of semisimple groups, volume 36 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, An overview based on examples.. 5, A q (λ) Sp(2, R) θ-stable q, K-

301 291 6, A q (λ), Jian-Shu Li [Li] Jian-Shu Li. Theta lifting for unitary representations with nonzero cohomology. Duke Math. J., 61(3): , 1990., Sp(2, R) Howe., Howe A q (λ). g- [Pr2] K- [ 2], Vogan A q (λ)., A q (λ) ( 19 ). 6 A q (λ),. (Sp, O) Howe [ ], [P]., [BW],., [KV], g, K. 1. K g. k := Lie K C g. 2. K g k Ad K (. ) 3. d(ad K (K)) = ad(k) ad(g), (g, K). 2. (h, L), (g, K) ι: (h, L) (g, K) ι alg ι gp ι = (ι alg, ι gp ) 1. ι alg l = d ι gp 2. ι alg Ad L (l) = Ad K (ι gp (l)) ι alg (l L). 3. (g, K), C(g, K) V. 1. V K- K-. 2. V g-, X g, k K, v V, k(xv) = (Ad(k)X)kv. 2

302 X k X g. K- V, V K = {v V dim C K v < } K-. 1 V K = V. 4. (g, K), U(g), R(K) K K-, R(g, K). R(g, K) := U(g) U(k) R(K) R(K) U(k) U(g) = U(g) C R(K)/ X w T X w T X U(g), w U(k), T R(K) R(g, K) K- R(g, K) C, K-. C(g, K) R(g, K). Hom g,k (V, W) = Hom R(g,K) (V, W) 5. ι: (h, L) (g, K), Z C(g, K), F = F h,l g,k : C(g, K) C(h, L) 1. F(V) = V 2. r v = ι(r) v r R(h, L). F (exact covariant functor). 6. P = P g,k h,l : C(h, L) C(g, K) 1. P(V) = R(g, K) R(h,L) V 2. r R(g, K), r(ω v) = (rω) v. P (right exact). 7. I = I g,k h,l : C(h, L) C(g, K) 1. I(V) = Hom R(h,L) (R(g, K), V) K 2. (rφ)(x) := φ(xr) φ I(V), r, x R(g, K). I (left exact). 8.. h g, P g,k h,k, Ig,K h,k. 9. V C(h, L) X j X j 1 X 0 V 0 ( ). P C(g, K) P(X j ) j 1 P(X j 1 ) P(X 0 ) 0, P P j P j (V) = Ker( j 1 )/ Im( j ). P 0 (V) = P(V). I, I j (V). 3

303 g g 0. g 0 C = g. g 0 θ : g 0 g 0 : g 0 g 0 R., (g, K) (reductive pair). 1. g Ad(K)-, ad(g)-. 2. Lie(K) = k 0 = g 0 (θ; 1)., p 0 = g 0 (θ; 1), g 0 = k 0 + p 0. k 0, p 0,. 3. p 0 k (g, K), h 0 g 0 θ-stable. θ(h 0 ) = h 0. ( ) q q = l + u. q, θq = q l h, θ-stable. θ-stable l l 0 g 0. L K l 0 k 0. (q, L K) 1,, (θ-stable). (l, L K). 12. t 0 = h k 0, a 0 = h p 0. (g, h), (k, t), (l, h). δ G, δ c, δ L, g, k, l. (u) u, δ(u). λ h, 1t 0 + a 0, Re(λ). 13. Z C(l, L K), Z # = Z C top u. (l, L K) top u C 2δ(u). Z L j, R j. L j (Z) := ( ) P g,k g,l K j Pg,L K q,l K F q,l K l,l K (Z# ) R j (Z) := ( I g,k g,l K ) j I g,l K q,l K Fq,L K l,l K (Z# ), q = l + ū (opposite).,. L j (Z), R j (Z): C(l, L K) C(g, K) 14. [KV, 5.35]. S := dim C (u k). j > S, L j (Z) = R j (Z) = [Kn, VIII.5]. W G. U(g) Z(g) Harish-Chandra γ g : Z(g) U(h) W G, U(h) W G. χ: Z(g) C, λ h /W G, χ(z) = χ λ (z) = λ(γ g (z)) (z Z(g)). V C(g, K) Z(g) χ λ, V χ λ ( λ). 16. g 0 G L G,, 4

304 294 (5.7) L G. 17. [KV, 5.25]. (5.7). Z C(l, L K) λ, L j (Z), R j (Z) C(g, K) λ + δ(u). 18. [KV, 5.99]. (5.7). Z C(l, L K) λ, admissible. Re λ + δ(u) α 0 (α (u)), 0 j < S, L j (Z) = R j (Z) = 0, ( ) L S (Z) R S (Z). 19. [KV, 8.2]. (5.7). Z C(l, L K) λ admissible (l, L K). λ Re λ + δ(u) α > 0 (α (u)), L S (Z) R S (Z), (g, K)-. 3 A q (λ) 2,. G, K., t 0 k 0, h t θ-stable., (g, K) (q, L K). (u) (g, h)., (5.7). 20. Z C(l, L K), λ h λ (l, h) = 0 1 (l, L K) C λ. C λ λ + δ L., A q (λ) := L S (C λ ), A q (λ) := R S (C λ ). 21. [KV, 5.109]., 1. A q (λ) λ + δ G. 2. λ Re λ + δ(u) α > 0 (α (u)), A q (λ) A q (λ). 22. (g, K) V, K- V K K-. µ 1t 0 K- K-, µ + 2δ c µ + 2δ c,. 23. [KV, 9.70,10.24]., 5

305 λ t0, λ a0,. 2. A q (λ) K- λ (k, t), λ + 2δ(u p) + n α α (n α Z 0 ) α (u p). K- λ + 2δ(u p) K- u p. q, q u p A q (λ) A q (λ). 4, 3. G. 25. G D K- D K (g, K). v, w D K, x G x v, w G, D. (g, K) D K. 26. [Kn, 12.20]. (g, K) G. h = t , h = t,. 28. [Kn, 9.20]. µ ( 1h). µ > 0. Λ = µ + δ G 2δ c K. µ,, D µ. 1. D µ µ. 2. D µ K- Λ D µ K- Λ + n α α (n α Z 0 ). α (p), α>0 29. W G, W K G, K. D µ D µ µ = wµ w W K. 15,, µ W G /W K. 30. D µ, µ Harish-Chandra., Λ Blattner., V C(g, K) HC = HC(V), K- BL = BL(V). HC(D µ ) = µ, BL(D µ ) = Λ., HC(A q (λ)) = λ + δ G (21 ), BL(A q (λ)) = λ + 2δ(u p) (23 ). 6

306 cos θ sin θ. : G = SL(2, R). K = S O(2) τ k ( sin θ cos θ ) = eikθ. δ G = 1. µ W G /{1} = 2. k 2 D + k 1 = τ k+2l, D k 1 = τ k 2l, l Z 0 l Z 0 0 τ k τ k+2 D + k 1 HC(D ± k 1 ) = ±(k 1), BL(D± k 1 ) = ±k. ( (k 1) (k 1) 28 HC.) 32. : G = Sp(2, R). K U(2). rank(g) = rank(k) G. W G /W K = 4. G C 2 - {±(e 1 ± e 2 ), ±2e 1, ±2e 2 }. e 1 e 2. (k 1, k 2 ) k 1 e 1 + k 2 e 2 R 2 ( 1h). Λ = (Λ 1, Λ 2 ) K- sym Λ 1 Λ 2 det Λ 2. τ k+4 (I). µ = (µ 1, µ 2 ) I, δ G = (2, 1), δ c = (1/2, 1/2) Λ = (µ 1 + 1, µ 2 + 2). (II). µ = (µ 1, µ 2 ) II, δ G = (2, 1), Λ = (µ 1 + 1, µ 2 ). regular dominant integral I (III). µ = (µ 1, µ 2 ) III, δ G = (1, 2), Λ = (µ 1, µ 2 1). (IV). µ = (µ 1, µ 2 ) IV, δ G = ( 1, 2), Λ = (µ 1 2, µ 2 II 1). IV III 33. [KV, ]., A q (λ) q b, h = l (g, K). A b (λ). 5 Sp(2, R) G = Sp(2, R)., h = t. 32 ( 1h) R (2), (k, h). 34. θ-stable q. α (g, h), g α. (k, h) ξ ( 1h) q. q = l + u ( ) l = h + g α, u = α (g,h), α,ξ =0 α (g,h), α,ξ >0 ξ, q. ξ = 0, q = g Sp(2, R) θ-stable q = l + u. g α 7

307 q = Q 3,0 (0) L = K, l = k u = g (2,0) + g (1,1) + g (0,2) = p + δ G = (2, 1) δ(u p) = ( 3 2, 3 2 ) ξ 36. q = Q 3,0 L = T, l = t u = p + + g (1, 1) δ G = (2, 1) δ(u p) = ( 3 2, 3 2 ) ξ 37. q = Q 2,0 l = t + g (0,2) + g (0, 2) L U(1) Sp(1, R) u = g (2,0) + g (1,1) + g (1, 1) δ G = (2, 1) δ(u p) = ( 3 2, 1 2 ) ξ 38. q = Q 2,1 L = T, l = t u = g (2,0) + g (1,1) δ G = (2, 1) δ(u p) = ( 3 2, 1 2 ) ξ + g (0,2) + g (1, 1) q = Q 1,1 39. l = t + g (1,1) + g ( 1, 1) L U(1) SL(2, R) δ G = (2, 1) δ(u p) = (1, 1) ξ u = g (2,0) + g (1, 1) + g (0, 2) 40. q = Q 1,2 L = T, l = t δ G = (1, 2) u = g (2,0) + g (1, 1) + g (0, 2) + g ( 1, 1) δ(u p) = ( 1 2, 3 2 ) ξ 41. q = Q 0,2 l = t + g (2,0) + g ( 2,0) L Sp(1, R) U(1) δ G = ( 1, 2) δ(u p) = ( 1 2, 3 2 ) ξ u = g (0, 2) + g (1, 1) + g ( 1, 1) 42. q = Q 0,3 L = T, l = t δ G = ( 1, 2) u = p + g (1, 1) δ(u p) = ( 3 2, 3 2 ) ξ 8

308 q = Q 0,3 (0) L = K, l = k δ G = ( 1, 2) u = g (0, 2) + g ( 1, 1) + g ( 2,0) = p δ(u p) = ( 3 2, 3 2 ) ξ 44. q = Q 0,0, l = g, L = G ξ 45., Sp(2, R) A q (λ). λ := AQ(A q (λ)). C λ L, λ (l, h) = 0 (20 ). λ θ-stable q A q (λ) ( ), K- ( ) K- ( ), 46 49,., 24, q = Q 3,0 (0) q = Q3,0, q = Q 0,3 (0) q = Q 0,3 A q (λ)., q = Q 0,0 = g. q = Q i, j Q j,i, R 2 (x, y) ( y, x) (46,47,48 ) q = Q 3,0, AQ : λ = (λ 1, λ 2 ), λ 1 λ 2 0 HC : λ + (2, 1) BL : λ + (3, 3) A q (λ) : 47. q = Q 2,0, AQ : λ = (λ 1, λ 1 ), λ 1 1 HC : λ + (2, 1) BL : λ + (3, 1) A q (λ) : 48. q = Q 2,1 AQ : λ = (λ 1, λ 2 ), λ 1 λ 2 0 HC : λ + (2, 1) BL : λ + (3, 1) A q (λ) : 9

309 q = Q 1,1, AQ : λ = (λ 1, λ 1 ), λ 1 1 HC : λ + (2, 1) BL : λ + (2, 2) A q (λ) : non-tempered 6 Sp(2, R) Howe 50. Howe. Howe [ 2]. G 1, G 2 Sp dual pair, (ω, P) Sp Fock (=Weil (g, K)- ). R(g i, K i, P) = {V : (g i, K i ) V P }, Howe V 2 = θ(v 1 ). θ : R(g 1, K 1, P) V 1 V 2 R(g 2, K 2, P): by V 1 V 2 R(g 1 + g 2, K 1 K 2, P), Sp(2, R) Howe. even case, G 1 = O(4), O(2), O(2, 2), G 2 = Sp(2, R), dual pair., S O 0 (3, 2) Sp(2, R) G 1 = SL(2, R), G 2 = O(3, 2) dual pair., [Li, 6.2]. 51. G 1 = O(4), G 2 = Sp(2, R). V 1. V 2 = θ(v 1 ). V 1 of O(4) V 2 of Sp(2, R) HC (a 1, a 2 ) (a 1, a 2 ) (a 1 > a 2 1) BL (a 1 1, a 2 ) (a 1 + 1, a 2 + 2) AQ (a 1 2, a 2 1) q Q 3,0 52. G 1 = O(2), G 2 = Sp(2, R). V 1. V 2 10

310 300. V 1 of O(2) V 2 of Sp(2, R) HC a 1 (a 1, 1) (a 1 2) BL a 1 (a 1 + 1, 1) AQ (a 1 2, 0) q Q 2,0 53. G 1 = O(2, 2), G 2 = Sp(2, R). V 1 HC (a, b) K- 4 Ind K 1 K τ 1 a+1 τ b (31 ). V 2 = θ(v 1 ). V 1 of O(2, 2) V 2 of Sp(2, R) HC (a, b) (a, b) (a > b 1) BL (a + 1, b) (a + 1, b) AQ (a 2, 1 b) q Q 2,1 54. G 1 = SL(2, R), G 2 = Sp(2, R) S O 0 (3, 2) O(3, 2). V 1 genuine, V 2 = θ(v 1 ), non-tempered A q (λ). V 1 of SL(2, R) V 2 of Sp(2, R) HC a 1/2 (a, a + 1) (a 2) BL a + 1/2 (a, a) AQ (a 2, 2 a) q Q 1,1 V 1 of SL(2, R) V 2 of Sp(2, R) HC b + 1/2 (b, b 1) (b 2) BL b 1/2 (b + 1, b + 1) AQ (b 2, b 2) q Q 3,0 (0) [ ] Reductive Dual Pair Weil 4, p.63 88, [ 1] Howe Duality 4, p , [ 2], Weil Howe duality,. [BW] A. Borel and N. Wallach. Continuous cohomology, discrete subgroups, and representations of reductive groups, volume 67 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, [Kn] Anthony W. Knapp. Representation theory of semisimple groups, volume 36 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, An overview based on examples. [KV] Anthony W. Knapp and David A. Vogan, Jr. Cohomological induction and unitary representations, volume 45 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, [Li] Jian-Shu Li. Theta lifting for unitary representations with nonzero cohomology. Duke Math. J., 61(3): ,

311 301 [M] C. Mœglin. Correspondance de Howe pour les paires reductives duales: quelques calculs dans le cas archimédien. J. Funct. Anal., 85(1):1 85, [P] Annegret Paul. On the Howe correspondence for symplectic-orthogonal dual pairs. J. Funct. Anal., 228(2): , [Pr1] Tomasz Przebinda. The oscillator duality correspondence for the pair O(2, 2), Sp(2, R). Mem. Amer. Math. Soc., 79(403):x+105, [Pr2] Tomasz Przebinda. The duality correspondence of infinitesimal characters. Colloq. Math., 70(1):93 102, [VZ] David A. Vogan, Jr. and Gregg J. Zuckerman. Unitary representations with nonzero cohomology. Compositio Math., 53(1):51 90,

312 302 Siegel ( ) k 2k 2 2k 4 k [HI 05] 2 Kohnen plus 1 4 (k ) (cf. [Ik 06]) Duke-Imamoglu- - ( )

313 [Gu 11a] 2 H. Saito N. Kurokawa[Ku 78] H. Maass A. Andrianov D. Zagier (cf. [EZ 85]) 1 k S 2k 2 (SL(2, Z)) = S + (Γ (1) k 1 0 (4)) = J 2 (1) cusp k,1 = Sk Maass (Sp(2, Z)) S + (Γ (1) (1) cusp k 1 0 (4)) k Kohnen J k, k ( 1) Sk Maass (Sp(2, Z)) [Sk 11] [Tk 11] [Ib 11] ( k ) 2 F S k (Sp(2, Z)) 2 F (Z) = N A(N)exp(2πitr(NZ)) F F ( ) n r 2 r 2 m (( )) n r A 2 r = (( nm )) d k 1 r A d 2 2d r. 2 m d (n,m,r) 2d 1 k Sk Maass (Sp(2, Z)) 2 l V l U l l l 2 ( V l U l [EZ 85, p.41] [Ib 11] 2

314 304 [Su 11] [Ao 11] ) M k (Sp(2, Z)) S k (Sp(2, Z)) k 2 F M k (Sp(2, Z)) k 2 F ( τ z z ω ) = m ψ m (τ, z)q m q := e 2πimω ϕ m k m V l 1.1. F m ψ m = ψ 1 V m 1.2. F m p ψ m V p = ψ mp + p k 1 ϕ m p U p. 1.1 V m V m V n = d (m,n) dk 1 V mn U d 2 d ( [EZ 85, p.43] ( [EZ 85, p.43] (10) U l V l ) V n, U d ) l T (l) S k (SL(2, Z)) 1.3. ϕ k m (1) ϕ(τ, 0) S k (SL(2, Z)) (2) l (ϕ V l )(τ, 0) = (ϕ(τ, 0)) k T (l). 1.2 Pullback 2 pullback H n n 3

315 305 F M k (Sp(2, Z)) k 2 F H 1 H 1 H 2 pullback F ( τ 0 ω 0 ) τ ω ω F ( τ 0 ω 0 ) τ F ( τ 0 0 ω ) = i u i (τ)f i (ω) (1.1) {f i } i M k (SL(2, Z)) {f i } i {u i } i F u i k ( ) F ( τ z z ω ) = F ( ω z z τ ) pullback M k (Sp(2Z)) (M k (SL(2, Z)) M k (SL(2, Z))) sym F ( τ 0 0 ω ) = i,j C i,j (F ) f i (τ)f j (ω) (C i,j (F )) i,j F (2) m p ψ m (τ, 0) T (p) = ψ mp (τ, 0) + p k 1 ψ m (τ, 0) p q m m F ( τ 0 ω 0 ) τ T (p) = (ψ m (τ, 0) T (p))q m m = m (ψ mp (τ, 0) + p k 1 ψ m (τ, 0))qm p = F ( τ 0 0 ω ) ωt (p) F ( τ 0 0 ω ) τ T (p) F ( τ 0 0 ω ) τ T (p) F ( τ 0 0 ω ) ωt (p) F ( τ 0 0 ω ) ω T (p) ( ( m a m q m ) k T (p) = m (a mp + p k 1 a m p )qm ) i (u i T (p)) (τ) f i (ω) = F ( τ 0 0 ω ) τ T (p) = i u i (τ)a i (p)f i (ω) 4

316 306 a i (p) f i p p u i T (p) = a i (p)f i u i = C i,i (F )f i F F ( τ 0 0 ω ) = i C i,i (F )f i (τ)f j (ω) ( F 2 ) pullback M Maass k (Sp(2, Z)) (M k (SL(2, Z)) M k (SL(2, Z))) diag (1.2) (M k (SL(2, Z)) M k (SL(2, Z))) diag M k (SL(2, Z)) M k (SL(2, Z)) i,j C i,jf i f j (C i,j ) i,j Remark. M k (SL(2, Z)) M k (SL(2, Z)) 1. (1.2) ( [Ic 05, Lemma1.1] ) 2. F C i,i (F ) 6 L- ( [Ic 05, T heorem2.1] ) 3. F M k (Sp(2, Z)) pullback (M k (SL(2, Z)) M k (SL(2, Z))) diag F F (( τ z z ω )) z (τ, ω ) F (Heim[He 10] ) Γ (n) 0 (4) Sp(n, Z) M 2n,2n(Z) Γ (n) 0 (4) := {( C A D B ) C 0 n mod 4} M + (Γ (n) k 1 0 (4)) n 2 5

317 307 M + (Γ (n) k 1 0 (4)) n k k n k 1 J (n) k,1 (4)) ( Eicher-Zagier[EZ 85] n = 1 M + k 1 2 (Γ (n) 0 [Ib 92] n > 1 [Tk 99] [Tk 11] ) - [HI 05] 1 ([HI 05]). k f g 2k 2 2k 4 F f,g S + (Γ (2) k 1 0 (4)) 2 L- L(s, F f,g ) = L(s, f)l(s 1, g) L- Zhuravlev [Zh 84] L- F n {α ± i,p } F p-parameter L(s, F ) := p n i=1 {( 1 α i,p p s+k 3/2) ( 1 α 1 i,p p s+k 3/2)} 1 p F N 1 1 p L(s, f) L(s, g) f g L- ( 2 [HI 02] 2 [Ib 08] 2 Nebentype ( ) 1 Haupttype ( ) 1 [Ib 08] ) 6

318 k 1 f g F f,g S + (Γ (2) k 1 0 (4)) 2 F f,g 0 F Zhuravlev L- 1 L(s, F f,g ) = L(s 1, g) L(s, f) 3 F f,g 2.1 F f,g k g S 2k 4 (SL(2, Z)) g = b(m) q m m b(1) = 1 g I(g) S k (Sp(4, Z)) (k ) I(g) 4 I(g) 1 I 1 (g) I(g) ( τ z t z ω ) = m Z >0 (I m (g)(τ, z))e 2πimω ( τ H 3, ω H 1, z M 3,1 (C)) I(g) I m (g) 3 k m I 1 (g) k I 1(g) G (3) cusp g S 2k 4 (SL(2, Z)) I(g) S k (Sp(4, Z)) I 1 (g) J k,1 G S + (Γ (3) k 1 0 (4)) 2 (3) cusp J k,1 k 1 3 Remark. I 1 (g) I 1 (g) ([Ha 11a] ) S k (Sp(n, Z)) 1st n = 2 2 S k (Sp(2, Z)) Kohnen-Skoruppa [KS 89] 1st 2 L- 7

319 309 H 2 H 1 H 3 G pullback G ( τ 0 0 ω ) = h F h,,g (τ)h(ω) (3.1) τ H 2 ω H 1 h S + k 1 2 (Γ (1) 0 (4)) g S 2k 4 (SL(2, Z)) G S + (Γ (3) k 1 0 (4)) 2 f S 2k 2 (SL(2, Z)) f ˆf S + (Γ (1) k 1 0 (4)) 2 (3.1) G F f,g := F ˆf,g F f,g S + (Γ (2) k 1 0 (4)) G 2 k S 2k 2 (SL(2, Z)) S 2k 4 (SL(2, Z)) S + (Γ (2) k 1 0 (4)) ((f, g) F f,g) 2 Remark. S 2k 2 (SL(2, Z)) S 2k 4 (SL(2, Z)) Kohnen S + (Γ (1) k 1 0 (4)) S+ (Γ (1) 2 k 3 0 (4)) S+ (Γ (2) 2 k 1 0 (4)) 2 Zagier [HI 05] [HI 05] 2.1 (1) F f,g (2) F f,g L- f g L- (1) (2) 4 G Fourier-Jacobi G Fourier-Jacobi 8

320 G Fourier-Jacobi k G S + (Γ (3) k 1 0 (4)) 3 G 2 g S 2k 4 (SL(2, Z)) Fourier-Jacobi Eichler-Zagier- 2.1 G Fourier-Jacobi G (( τ z t z ω )) = ϕ m (τ, z)q m, m 0,3 mod 4 τ H 2, ω H 1, z M 2,1 (C) G Fourier m m 0, 3 mod 4 ϕ m k 1 2 m Remark. ϕ m G Sp(2, Z) Γ (2) 0 (4) G ϕ m ϕ m V l J (2) k,m 2 k m p i = 1, 2 V (2) p,i : J (2) (2) J k 1 2,m k 1 2,mp2 V (2) p,i (cf. [Ha 11b].) V (2) p,i V (2) p,1 V (2) p,2 diag(1, p, p2, p) diag(1, 1, p 2, p 2 ) 2 p = 2 V (2) 2,i J (2) k 1 2,m 9

321 311 U (2) l : J (2) (2) J k 1 2,m k 1 2,ml2 ϕ J (2) ϕ(τ, z) ϕ(τ, lz) ( k 1 2,m τ H 2 z M 2,1 (C).) 4.1. p m ( ) ϕ m V (2) m p,1 = p b(p) ϕ m U p + ϕ mp 2 + p k 2 ϕ m U p + p 2k 3 ϕ m p U p 2 p 2, ϕ m V (2) p,2 = (p2k 4 p 2k 6 ) ϕ m U p ( ( m +b(p) ϕ mp 2 + p k 2 p ) ϕ m U p + p 2k 3 ϕ m p 2 U p 2 ). b(p) g S 2k 4 (SL(2, Z)) p m ( ) p 2 ϕ m 0 p 2 p p = 2 ( t t 2) = ( 1) (t 1 mod 2 ) ( t 2) = 0 (t 0 mod 2 ) Remark b(p) g p p 2 [Yz 86, Yz 89] Kohnen [Ko 02] Kohnen- [KK 05] [Yn 10] [Ha 11a] 2 [Tn 86] ) ( 4.1) ) 2.1 T ( 1 p p 2 p T ( 1 1 p 2 p 2 ) ( 1 p p 2 p S + (Γ (2) k 1 0 (4)) ( 1 1 p 2 p 2 )

322 ϕ m G m ϕ m k 1 2 m (1) ϕ m (τ, 0) S + (Γ (2) k 1 0 (4)) 2 (2) p (ϕ m V (2) p,1 )(τ, 0) = (ϕ m(τ, 0)) T ( 1 p p 2 p ) (ϕ m V (2) p,2 )(τ, 0) = (ϕ m(τ, 0)) T ( 1 1 p 2 p 2 ) (ϕ m (τ, 0)) T ( 1 p p 2 p = p b(p) ϕ m (τ, 0) + (( τ 0 G 0 ω)) τ ) = (ϕ m V (2) p,1 )(τ, 0) ( ϕ p2 m(τ, 0) + T ( 1 p p 2 p = p b(p) ϕ m (τ, 0)q m m + ( ( m ϕ p 2 m(τ, 0) + p m (( )) τ 0 = p b(p) G + G 0 ω ) ( m p = m ) p k 2 ϕ m (τ, 0) + p 2k 3 ϕ m p 2 (τ, 0) ) ( ϕ m (τ, 0) T ( 1 p p 2 p ) p k 2 ϕ m (τ, 0) + p 2k 3 ϕ m (( τ 0 0 ω)) ω T (p 2 ). )) p 2 (τ, 0) ) G (( τ 0 ω 0 )) τ G (( τ 0 ω 0 )) ω G (( τ 0 ω 0 )) τ ω ( 2 1 ) T (p 2 ) Kohnen h(ω) = m c mq m S + (Γ (1) k 1 0 (4)) 2 (h T (p 2 ))(ω) := m ( c mp 2 + ( ( 1) k+1 m p q m q m ) ) p k 2 c m + p 2k 3 c m q m p 2 11

323 313 G (( τ 0 0 ω )) = h F h,g (τ)h(ω) F f,g = F ˆf,g F f,g T ( 1 p p 2 p ) = (p b(p) + a(p))f f,g a(p) ˆf T (p 2 ) f p 4.1 F f,g T ( 1 1 p 2 p 2 ) = (p 2k 4 p 2k 6 + a(p)b(p))f f,g F f,g λ 1 (p) = p b(p)+a(p) λ 2 (p) = p 2k 4 p 2k 6 +a(p)b(p) F f,g F f,g L- L(s, F f,g ) = ( 1 λ1 (p)p s + (pλ 2 (p) + p 2k 5 (1 + p 2 ))p 2s λ 1 (p)p 2k 3 3s + p 4k 6 4s) 1 p = p {( 1 b(p) p 1 s p 2k 3 2s) ( 1 a(p) p s p 2k 3 2s)} 1 = L(s 1, g) L(s, f) 4.1 ( 2.1) k 1 2. k M k 1/2 det(2m) S. Böcherer 5. 12

324 I 1 (g)(τ, z) k (τ, z) H 3 M 3,1 (C) ( τ z t z ω ) H 4 (τ H 3 ω H 1 z M 3,1 (C)) I 1 (g)(τ, z)e 2πiω = ϕ M (τ, z )e 2πiTr(Mω ) M Sym 2 ϕ M 2 k M ( ) Sym τ 2 2 ( z t z ω ) = τ z (τ H t z ω 2 ω H 2 z M 2,2 (C)) M M = ( 1 ) (3) cusp ( I 1 (g) J k,1 ) (3) cusp J k,1 M Sym 2 M=( 1 ) (2) cusp J k,m (2) cusp J k,m 2 k M ( Ziegler [Zi 89] ) Remark. ϕ M 3 I(g) (g 4 ) M I(g) S k (Sp(4, Z)) 1st F-J (3) cusp I 1 (g) J k,1 E-Z-I F-J Ikeda lift F-J {ϕ M } M (2) cusp J k,m ι {ϕ m } m M Sym 2 M=( 1 ) g S 2k 4 (SL 2 (Z)) 13 G S + k 1 2 (Γ (3) 0 (4)) (2) cusp J k 1 2,m m 0,3 mod 4

325 315 (2) cusp J k 1 2,m 2 k 1 2 m ι 5.2 {ϕ m } m {ϕ M } M 5.2 ( ) ( ) Eichler-Zagier- 1 ( ) ( ) ( ) ( ) ( ) ι (2) cusp J p = 2 J k 1 2,m p2 V (2) p,i (i = 1, 2) U p (2) ( (2) cusp k 1 2,m (2) cusp ) J k,m M[( ) p ] ( A[B] := t BAB ) V (2) ( ) p 0,i 0 1 (i = 1, 2) U (2) ( p ) V ( (2) p U ( (2) p ),i (2) cusp : J k,m (2) cusp ) : J k,m J (2) cusp ( ) k,m[ p 0 ] 0 1 J (2) cusp ( k,m[ p 0 )]. 0 1 (i = 1, 2), ϕ M ϕ m ϕ M V (2) ( p ),i ϕ m V (2) p,i ϕ M U (2) J (2) cusp J k,m (2) cusp ( k,m[ p (2) cusp J k 1 2,m (2) cusp ) J ] k 1 2,mp2 ( p ) ϕ m U (2) p 4.1 {ϕ m } m {ϕ M } M

326 316 Remark. p = [Ik 01] ϕ M I(g) ϕ M 4 M 5.2 V (2) ( ) p 0,i 0 1 (i = 1, 2) U (2) ( p ) ( k ) 5.4 M = ( 1 ) Sym 2 Boecherer [Bo 83, Satz 7] n M = ( 1 ) Sym 2 2 m = det(2m) D 0 Q( m) f := m D 0 m 0, 3 mod 4 f h k 1 2 (m) k 1 2 Cohen ([Co 75] ) m- 15

327 317 ([Co 75, p.272] H(k 1, m) ) h k 1 2 (m) = h k 1 2 ( D 0 ) m k 3 2 µ(d) d f ( D0 d ) ( ) f d 1 k σ 3 2k d σ l (a) = a l µ d a g k g k (m) := ( m ) ( ( ) ) µ(d) h k 1 2 d 2 g k (p 2 m) = p 2k 3 m p p k 2 g k (m) d f ( ) m p (k p ( ) ) m p e (n 2) k,m n k M E (n 2) k,m n 2 k M ( [Zi 89, Theorem 2.1] ) 5.1. k > n + 1 e (n 2) k,m (τ, z) = d f g k ( m d 2 ) E (n 2) k,m[ t W d 1 ] (τ, zw d) d det(w d ) = d W 1 d Mt W 1 d = ( 1 ) Sym 2 W d M 2,2 (Z) W d [Bo 83, Satz 7] ( e (2) k,m ) 4.1 e (2) k,m E(2) k,m V (2) ),i ( p E (2) k,m V ( (2) p (2) V ( p ),i (i = 1, 2) E(2) k,m ),i M m D 0 f 5.4 n n 16

328 318 E (n) k := γ Γ 0 \Γ 1 k γ n (cf. [Gu 11b].) ( Γ := Sp(n, Z), Γ 0 := {( ) } A B 0 n D Γ ) α M 2n (Z) GL + 2n (Q) M 0 = {( ) } A B 0 n D ΓαΓ ΓαΓ E (n) k ΓαΓ = 1 k γγ = 1 k γ γ Γ\ΓαΓ γ Γ 0 \Γ = δ Γ 0 \M 0 γ Γ 0 \Γ 1 k δγ = c δ,k E (n) k E (2) k,m E(n) k V (2) ( p γ Γ 0 \ΓαΓ δ Γ 0 \M 0 c δ,k γ Γ 0 \Γ V (2) E (2) k,m V (2) E (2) (2) k,m V ( p (2) U ( p k γ ),i ( p ),i ),i ( ) p 0,i E(2) ( ) k,m[ p 0 ] E(2) k,m ) E (2) (2) k,m[x ( p U 1 ) 1 ] ( p 1 X( p )t 1 ) ( p f X 1 M[X 1 ( p 1 ) 1 ] ) p f E (2) (2) k,m V ( p 0 ),1 = 0 1 ( p 2k+3 + p 4k+8 + p 4k+7 p 3k+5 ( m p ( ( m + p 2k+4 + p 3k+5 p )) E (2) k,m[ ( p 0 )]. 0 1 )) E (2) k,m (2) U ( ) p G 2,l p (A) 2 A Sym 2 l 2 G 2,l p (A) := x = t x M 2,2 (Z/pZ) rank p (x )=l ( ) 1 e p tr(ax ). [Sa 91] 2 p [Sa 91] 17

329 319 ( [Sa 91, Proposition 1.9 (1)] q n (t+1) 2 t+1 n q 2 ) A 2 E (2) (2) k,m V ( p 0 ),i 0 1 [Ha 11b] Remark. p = 2 E (2) (2) k,m V ( p ),i 5.1 e (2) k,m e (2) (2) k,m U ( p 0 (2) V ( p ),i e(2) ( k,m[ p 0 ) e (2) (2) k,m[x ( p U 1 ) 1 ] ( p 1 X( p )t 1 ) k > 5 k )] [Ao 11] : Borcherds product, [Bo 83] S. Boecherer: Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. I, Math.Z. 183 (1983), [Co 75] H. Cohen: Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), [EZ 85] M. Eichler and D. Zagier: Theory of Jacobi Forms, Progress in Math. 55, Birkhäuser, Boston-Basel-Stuttgart, (1985). [Ha 11a] S. Hayashida: Fourier-Jacobi expansion and the Ikeda lift, Abh. Math. Sem. Univ. Hamburg 81 no.1 (2011), [Ha 11b] S. Hayashida: Maass relations for generalized Cohen-Eisenstein series of degree two and of degree three, preprint. [HI 02] S. Hayashida and T. Ibukiyama: Siegel modular forms of half integral weights and a lifting conjecture. (Automorphic forms, automorphic representations and automorphic L-functions over algebraic groups), No (2000), [HI 05] S. Hayashida and T. Ibukiyama: Siegel modular forms of half integral weights 18

330 320 and a lifting conjecture, Journal of Kyoto Univ. 45 no.3 (2005) [He 10] B. Heim: On the Spezialschar of Maass, Int. J. Math. Math. Sci. (2010) 15pp. [Gu 11a] : Siegel Hecke, [Gu 11b] : Siegel Eisenstein Fourier, [Ib 92] T. Ibukiyama: On Jacobi forms and Siegel modular forms of half integral weights, Comment. Math. Univ. St. Paul. 41 no.2 (1992), [Ib 08] T. Ibukiyama: A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder s conjecture on congruences, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, (2008) [Ib 11] : Saito Kurokawa for level N, [Ic 05] A. Ichino: Pullbacks of Saito-Kurokawa lifts. Invent. Math. 162 no. 3, (2005), [Ik 01] T. Ikeda: On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math. (2) 154 no.3, (2001), [Ik 06] T. Ikeda: Pullback of the lifting of elliptic cusp forms and Miyawaki s conjecture, Duke Math. J. 131 no.3, (2006), [Ko 02] W. Kohnen : Lifting modular forms of half-integral weight to Siegel modular forms of even genus, Math, Ann. 322 (2002), [KK 05] W. Kohnen and H. Kojima : A Maass space in higher genus, Compos. Math. 141 No.2, (2005), [KS 89] W. Kohnen and N. -P. Skoruppa: A certain Dirichlet series attached to Siegel modular forms of degree two, Invent. Math. 95 (1989), [Ku 78] N. Kurokawa: Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two. Invent. Math. 49 (1978), [Sa 91] H. Saito: A generalization of Gauss sums and its applications to Siegel modular forms and L-functions associated with the vector space of quadratic forms, J. Reine Angew. Math. 416 (1991), [Sk 11] : Shimura, [Su 11] : Oda Lift, [Tk 99] K. Takase: On Siegel modular forms of half-integral weights and Jacobi forms. Trans. Amer. Math. Soc. 351 No. 2, (1999), [Tk 11] : Siegel Jacobi, [Tn 86] Y. Tanigawa: Modular descent of Siegel modular forms of half integral weight 19

331 321 and an analogy of the Maass relation. Nagoya Math. J. 102 (1986), [Yn 10] S. Yamana: Maass relations in higher genus, Math. Z. 265 No.2, (2010), [Yz 86] T. Yamazaki: Jacobi forms and a Maass relation for Eisenstein series, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 33 (1986), [Yz 89] T. Yamazaki: Jacobi forms and a Maass relation for Eisenstein series II, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 36 (1989), [Zh 84] V. G. Zhuravlev: Euler expansions of theta transforms of Siegel modular forms of half-integral weight and their analytic properties, Math. sbornik. 123 (165) (1984), [Zi 89] C. Ziegler: Jacobi forms of higher degree, Abh. Math. Sem. Univ. Hamburg. 59 (1989), [email protected] 20

332 322 On v-adic periods of t-motives ( 1 ) t M v M v M 19 (2011 ) 1 Riemann Q Carlitz t Betti ( ) Papanikolas v v v t v Papanikolas 0 Riemann 1. n 2 tr.deg Q Q (π, ζ(2),..., ζ(n)) = n n/2 π x x x ζ(2n)/π 2n Q 1 π Q 1 Carlitz F q q p K := F q (θ) F q 1 n ζ C (n) := a F q [θ], a n K := F q ((θ 1 )) [email protected] 1

333 323 Carlitz F q [θ] K K Z Q R Riemann m n ζ C (p m n) = ζ C (n) pm π ( 2π 1 ) π π := θ( θ) 1 q 1 (1 θ 1 qi ) 1 K (( θ) 1 i=1 q 1 n ζ C (n)/ π n K Carlitz Z = 2 F q [θ] = q 1 Riemann K Chang Yu 2 ([CY, Corollary 4.6]). n 1 q 1 ) tr.deg K K ( π, ζ C (1),..., ζ C (n)) = n + 1 n/p n/(q 1) + n/p(q 1) Papanikolas t t K K t θ T := {f K [[t]] f t 1 } L := Frac(T) T L Frobenius σ F q (t) M t t K[t] Betti H B (M) L K[t] M H B (M) F q (t) M dim Fq (t) H B (M) = rank K[t] M =: r H B (M) x M m L K[t] M x m Ψ = (Ψ ij ) i,j GL r (L) M M H B F q (t) Γ Γ F q (t) Papanikolas 3 ([Pa, Theorem 4.3.1, ]). M dim H B (M) = rank M t Ψ Γ tr.deg K(t) K(t)(Ψ11, Ψ 12,..., Ψ rr ) = dim Γ v M v V (M) H B V 3 3 2

334 Papanikolas ABP-criterion ([ABP]) tr.deg K K(Ψ11 t=θ, Ψ 12 t=θ,..., Ψ rr t=θ ) = dim Γ Anderson Thakur [AT] t M Ψ t=θ ζ C Γ 2 v v ζ C,v ABP-criterion v Ψ ζ C,v 3 v ζ C,v 2 t 3 v t t t t 5. K t K F q [t] E d 0 K K E K = G d a N (t θ) N Lie(E) = 0 Hom gr.sch./k (E, G a ) K[t] t G d a F q[t] Z q Frobenius End(G d a) 6 (Carlitz ). C := G a,k F q [t] F q [t] End(C); t (x θx + x q ) C t Carlitz t Frobenius σ σ : K[t] K[t]; i a i t i i a q i ti 7. K t K[t] M ϕ: M M (M, ϕ) M K[t] ϕ: M M σ 3

335 325 det ϕ = c(t θ) n (c K, n 0) M K[ϕ] det ϕ M K[t] m ϕm = Am A ((K[t]) = K det ϕ 3 ) (M, ϕ) M t E M E := Hom gr.sch./k (E, G a ) E t M E t G a q M E ϕ M E K t 8. E M E K t K t 9 (Carlitz t ). C 6 Carlitz M C K[t] K[t] ϕ a M C ϕ(a) = (t θ)σ(a) Carlitz t 3 v Papanikolas v v F q [t] K K sep F q (t) v K(t) v K sep (t) v F q (t) K(t) K sep (t) v σ K(t) v K sep (t) v F q (t) v K t M v V (M) := ( K sep (t) v K[t] M ) σ ϕ ( ) σ ϕ V (M) F q (t) v M t v 10. K(t) v ϕ K(t) v N σ ϕ: N N N ϕ K(t) v ϕ N K(t) v ϕ K sep (t) v Fq (t) v V (N) K sep (t) v K(t)v N V (N) := (K sep (t) v K(t)v N) σ ϕ K(t) v ϕ C C V F q (t) v M K t K(t) v K[t] M C 4

336 326 M K t r := rank K[t] M r = dim Fq (t) v V (M) V (M) x M m K sep (t) v K[t] M x m Ψ = (Ψ ij ) i,j GL r (K sep (t) v ) Ψ GL r (K[t])\ GL r (K sep (t) v )/ GL r (F q (t) v ) well-defined Ψ Ψ ij M v v = d l=1 (t λ l) F q K sep (t) v = K sep ((t λ l )) l Ψ ij = (Ψ ijl ) l Γ v K(t) v K[t] M C ( ) Γ v F q (t) v 3 v 11 ([Mi, Theorem 4.14, 5.15]). M K t Ψ ijl Γ v l tr.deg K(t)v K(t) v (Ψ 11l, Ψ 12l,..., Ψ rrl ) = dim Γ v 12. C N V (N) K Galois G K := Gal(K sep /K) G K F q (t) v Rep Fq (t) v (G K ) V C Rep Fq (t) v (G K ) t M Γ v Im(G K GL(V (M))) Zariski M 9 Carlitz t Γ v = G m 1 [ABP] [AT] [CY] [Mi] [Pa] G. W. Anderson, W. D. Brownawell, M. A. Papanikolas, Determination of the algebraic relations among special Γ-values in positive characteristic, Ann. of Math. 160 (2004) G. W. Anderson, D. S. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. of Math. 132 (1990) C.-Y. Chang and J. Yu, Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math. 216 (2007) Y. Mishiba, On v-adic periods of t-motives, preprint, arxiv: M. A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math. 171 (2008)