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

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1 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

2 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 ) i j i,j C- 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

3 2.3 ( 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

4 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

5 [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

6 (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

7 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

8 4.3 (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

9 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

10 (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

11 ( [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

12 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

13 [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

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 c

$2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c$ 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] 2 2.1 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,