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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