k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+

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1 1 SL 2 (R) γ(z) = az + b cz + d ( ) a b z h, γ = SL c d 2 (R) h 4 N Γ 0 (N) {( ) } a b Γ 0 (N) = SL c d 2 (Z) c 0 mod N θ(z) θ(z) = q n2 q = e 2π 1z, z h n Z Γ 0 (4) j(γ, z) ( ) a b θ(γ(z)) = j(γ, z)θ(z) z h, γ = Γ c d 0 (4) ( ) ( ) 1 j(γ, z) 2 a b = (cz + d) γ = Γ d c d 0 (4) k > 0 f(z) Γ 0 (N) k + (1/2) f(γ(z)) = j(γ, z) 2k+1 f(z) γ Γ 0 (N)

2 k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+(1/2) (Γ 0 (N)) p N Hecke T k+(1/2) (p 2 )f = λ p f p N. Hecke T 2k (p) g S 2k (Γ 0 (M)) T 2k) (p)g = λ p g p N. 2 SL 2 (Q)\SL 2 (A Q ) SL 2 (Q)\SL 2 (A Q ) SL 2 (A Q ) SL 2 (A Q ) Siegel Sp n (A Q ) Sp n (A Q ) F SL 2 (F ) 2 SL 2 (F ) g SL 2 (F ) x(g) = { c d c 0 c = 0 c(g 1, g 2 ) = x(g 1) x(g 1 g 2 ), x(g 2 ) x(g 1 g 2 ) g 1, g 2 SL 2 (F )

3 , F Hilbert c(g 1, g 2 ) SL 2 (F ) SL 2 (F ) 2 SL 2 (F ) F C F SL 2 (o F ) o F F F 2 SL 2 (F ) SL 2 (F ) SL 2 (o F ) {( ) } a b Γ 0 (4) = SL c d 2 (o) c 4o F 4 SL 2 (A Q ) 2 SL 2 (A Q ) SL 2 (A Q ) v SL 2 (Q v ) SL 2 (A Q ) SL 2 (Q v ) SL 2 (A Q ) SL 2 (Q v ) SL 2 (A Q ) SL 2 (Q v ) SL 2 (A Q ) SL 2 (A Q ) SL 2 (Q) SL 2 (Q) SL 2 (A Q ) SL 2 (Q) SL 2 (A Q ) SL 2 (Q) SL 2 (A Q ) SL 2 (Q)\ SL 2 (A Q ) 3 Waldspurger-Shimura F ψ : F C SL 2 (F ) π SL 2 (F ) SL 2 (F ) {±1} genuine π SL 2 (F ) genuine π

4 PGL 2 (F ) τ = θ( π, θ) θ( π, ψ) (0) π ψ-whittaker model ξ F ψ ξ (x) = ψ(ξx), χ ξ (t) = ξ, t, x F, t F θ( π, ψ ξ ) χ ξ (0) ξ F θ( π, ψ ξ ) χ ξ ξ θ( π, ψ ξ ) χ ξ (0) Wald( π, ψ) (1) π τ = Wald( π, ψ) Wald( π, ψ) = τ SL 2 (F ) genuine π π (2) π τ = Wald( π, ψ) Wald( π, ψ) = τ SL 2 (F ) genuine π 2 π +, π π + = π F A F ψ : A/F C π = v π v SL 2 (A) π 1 τ = Wald( π, ψ) := v Wald( π v, ψ v ) PGL2(A) S π v F π Wald( π, ψ) = Wald( π, ψ) (1) v / S π = π (2) v / S τ = π +, π π = π ε v, ε v {±1} ε v = 1 v S (1), (2) SL 2 (A) π = v π v SL 2 (A) 4 2 SL 2 (A) SL 2 (A)

5 3 n > 2 SL 2 (Q) SL 2 (A Q ) n n p 1 mod n SL 2 (Q p ) n n F G G(A F ) G(A F ) G(F ) G F K K 2 (F ) K 2 (F ) = F F / x (1 x) x F, x 0, 1 x (1 x) x F, x 0, 1 {x (1 x) x F, x 0, 1} F F F F K 2 (F ) (x, y) x, y x, 1 x = 1 (x 0, 1) G F 1 K 2 (F ) E G(F ) 1 G Sp n (F ) F 0 F 1 µ(f ), 1 n µ n µ n µ(f ) F v n Hilbert F v F v µ n K 2 (F v ) µ n 1 K 2 (F v ) E v G(F v ) 1 K 2 (F v ) µ n push out 1 µ n G(F v ) G(F v ) 1

6 v n v G(F v ) G(F v ) G(o v ) v 1 µ n G(F v ) G(F v ) 1 1 µ n G(A F ) G(A F ) 1 G(A F ) G(A F ) 1 K 2 (F ) v µ(f v ) µ n 1 G(A F ) G(A F ) G(F ) G(F ) G(A F ) G(F ) G(A F ) G(F ) G(A F ) Brylinski-Deligne G F 0 F F G F G derived group G der G der G der ( F ) = [G( F ), G( F )] T F Y = Hom F (G m, T ) T cocharacter group Y Galois Gal( F /F ) S Zar F Zariski site K K 2, G S Zar K 2, G Brylinsky-Deligne [1] S Zar 1 K 2 E G 1 Z Y 2 Q Weyl Galois Gal( F /F )

7 5. SL 2 (A) n F n µ(f ), µ n Hilbert SL n (F ) n 2 2 (( )) a b x = c d 2 c(g 1, g 2 ) { c if c 0, d if c = 0. c(g 1, g 2 ) = x(g 1) x(g 1 g 2 ), x(g 2 ) x(g 1 g 2 ) SL 2 (F ) n [g 1, ζ 1 ], [g 2, ζ 2 ] SL 2 (F ) µ n [g 1, ζ 1 ] [g 2, ζ 2 ] = [g 1 g 2, ζ 1 ζ 2 c(g 1, g 2 )]. SL 2 (F ) SL 2 (F ) g SL 2 (F ) [g, 1] [g] SL 2 (F ) H H SL 2 (F ) H F n o SL 2 (o) SL 2 (o) s : SL 2 (o) SL 2 (o) s(sl 2 (o)) SL 2 (o) τ + : GL 2 (F ) SL 2 (F ) τ : GL 2 (F ) SL 2 (F ) τ + (g) = (det g) n/2 g n τ (g) = (det g) n/2 g n PGL 2 (F ) PGL 2 (F ) SL 2 (F ) τ + : PGL 2 (F ) SL 2 (F ), τ : PGL 2 (F ) SL 2 (F ) h SL 2 (F ) good Z SL2 (F )(h) = Z SL2 ([h]) (F ) Z SL2 (F )(h) h SL 2 (F ) Z SL2 ([h]) [h] SL (F ) 2 (F ) SL 2 (F ) h SL 2 (F )

8 good h = τ + (g) h = τ (g) g PGL 2 (F ) h SL 2 (F ) τ + g PGL 2 (F ) τ + (g) h SL 2 ( F ) h SL 2 (F ) τ g PGL 2 (F ) τ (g) h F ψ : F C x F Schwartz ϕ S(F ) F ϕ(t)ψ(xt 2 ) dt = α ψ (x) 2x 1/2 F ˆϕ(t)ψ( x 1 t 2 /4) dt, α ψ (x) C ˆϕ(t) ϕ Fourier ˆϕ(t) = F ϕ(u)ψ(tu) du α ψ (x) x Weil h SL 2 (F ) g GL 2 (F ) τ + δ + ψ ([h, ζ], g) δ + ψ ([h, ζ], g) = α ψ (1) ζ α ψ (det g) (det g)n/2, x(h) if n 2 mod 4, ζ (det g) n/2, x(h) if n 0 mod 4. h g τ + δ + ψ ([h, ζ], g) = 0 g g PGL 2 (F ) δ + ψ ([h, ζ], g) g PGL 2(F ), h = τ (g) SL 2 (F ) δ ψ ( h, g) := α ψ (1) 2 δ + ψ ([ 1 2] h, g). δ + ψ ([h, ζ], g), δ ψ ([h, ζ], g) [h, ζ] SL 2 (F ) C 0 (PGL 2 (F )) PGL 2 (F ) φ C 0 (PGL 2 (F )) g PGL 2 (F ) I(g, φ) = (g) φ(xgx 1 ) dx, PGL 2 (F )/Z PGL2 (F )(g) Z PGL2 (F )(g) g PGL 2 (F ) (g) Weyl

9 SL 2 (F ) φ anti-genuine φ(ζ h) = ζ 1 φ( h), ζ µ n C 0 ( SL 2 (F )) SL 2 (F ) anti-genuine φ C 0 ( SL 2 (F )) h = [h, ζ] SL 2 (F ) I( h, φ) = (h) φ( x h x 1 ) d x SL 2 (F )/Z ( h) SL2 (F ) h SL 2 (F ) good I( h, φ) = 0 φ + C 0 (PGL 2 ) φ C 0 ( SL 2 (F )) δ + ψ δ + ψ ([h], g)i([h], φ) = I(g, φ+ ), g PGL 2 (F ) h h SL 2 (F ) g τ + φ C 0 ( SL 2 (F )) δ + ψ φ C 0 (PGL 2 (F )) δ ψ ([h], g)i([h], φ) = I(g, φ ), g PGL 2 (F ) h φ +, φ n o Hecke PGL 2 (o) Hecke H( SL 2 //SL 2 (o)) φ +, φ F µ n µ(f ) n SL 2 (A) n SL 2 (A) A/F ψ : A/F C C 0 (PGL 2 (A)) PGL 2 (A) g = (g v ) PGL 2 (A) φ = v φ v C 0 (PGL 2 (A)) I(g, φ) = v I(g v, φ v ).

10 C 0 ( SL 2 (A)) SL 2 (A) anti-genuine SL 2 (A) φ anti-genuine φ(ζ h) = ζ 1 φ( h), ζ µ n h = (h v ) SL 2 (A), φ = v φ v C 0 ( SL 2 (A)) I(h, φ) = v I(h v, φ v ). φ C 0 ( SL 2 (A)) φ +, φ C 0 (PGL 2 (A)) h SL 2 (F )/ h: ell. reg. I(h, φ) = g PGL 2 (F )/ τ ± (g): ell. reg. ( I(g, φ + ) + I(g, φ ) ). h g τ ± SL 2 (A) PGL 2 (A) Wen-Wei Li [3] [1] J.-L. Brylinski and P. Deligne, Central extensions of reductive groups by K 2, Publ. Math. IHES 94 (2001), [2] P. Delinge, Extensions centrales de groupes algebriques simplement connexes et cohomologie galoisienne, Publ. Math. IHES 84 (1996), [3] Wen-Wei Li, La formule des traces pour les revêtements de groupes réductifs connexes, I, arxiv: , II. arxiv: , III. arxiv: , IV. arxiv: , [4] J. Milnor, Introduction to algebraic K-theory, Annals of Mathematics Studies, 72 Princeton University Press.

11 [5] G. Shimura, On modular forms of half integral weight Ann. of Math. (2) 97 (1973), [6] T. Shintani On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), [7] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulair de poid demi entier J. Math. Pures Appl. (9) 60 (1981), no. 4, [8] J.-L. Waldspurger, Correspondence de Shimura, J. Math. Pures Appl. (9) 59 (1980), no. 1, [9] J.-L. Waldspurger, Correspondances de Shimura et quaternions Forum Math. 3 (1991), no. 3, [10] A. Weil, Sur certains groupes d operateurs unitaires Acta Math. 111 (1964) [11] Weissman, Metaplectic Tori over Local Fields Pacific J. Math. 241 (2009), no. 1,

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,