Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p

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1 Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler p SL 2 (Z) Γ 0 (p) new forms) p 1

2 2 parahoric p mod p Borel Sp(2, Q) p parahoric ( Tits building middle parahoric parahoric Sp(2, Q) Extended Dynkin diagram folding middle parahoric [12], [13], [14] Ralf Schmidt 2. Compact p D p D O G G = {g M 2 (D); gg = n(g)1 2, n(g) Q } g = (g ij ) M 2 (D) g = (g ji ) g ij g ij main involution G G A v Q place G v G A v H G = {g M 2 (H); gg = n(g)1 2 } G 1 = {g G ; n(g) = 1} G 1 C = Sp(2, C) Sp(2, R) G p U p = M 2 (O p ) G p D 2 O 2 principal genus D 2

3 v Dv 2 G G v U p Op 2 G p principal genus [4] [6] q p U q = M 2 (O q ) G p q p G q = GSp(2, Qq ), U q = GSp(2, Qq ) U(p) = G U p G A G 1 ρ ρ(±1 2) = 1 dominant integral weight) (f 1, f 2 ) (f 1, f 2 Z, f 1 f 2 0) ρ(±1 2 ) = 1 f 1 f 2 mod 2 V ρ (ρ, V ) G A G A G G /{center} = G 1 /{±1 2 } GL(V ) ρ U(p) M ρ (U(p)) = M f1,f 2 (U(p)) = {f : G A V ; f(uga) = ρ(u)f(g)} G A U(p) ρ (f 1, f 2 ) z G A U(p)zU(p) M ρ (U(p)) U(p)zU(p) = d i=1 z iu(p) ([U(p)zU(p)]f)(g) = n q p d i=1 U q ρ(z i )f(z 1 i g) T (n) = {(z v ) G A ; v z v M 2 (O v ), n(z v ) nz v } n p q GSp(2, Q q ) v = p [11] Spinor L q p λ(q δ ) T (q δ ) 1 λ(q)q s +(λ(q) 2 λ(q 2 ) q f 1+f 2 +2 )q 2s λ(q)q f 1+f s +q 2f 1+2f s. p L V [11] H 2 = R 8 H 2 [10] H 2 G 3

4 4 multiplicity free H 1 = {a H; n(a) = 1} H 2 H 1 G 1 H2 H 1 G 1 multiplicity free ( ρ V G A = h i=1 U(p)g ig Γ i = G g 1 i U(p)g i V Γ i = {v V ; ρ(γ)v = v for all γ Γ i } M ρ (U(p)) h i=1v Γ i principal genus h 1 [8] 3. ρ GL 2 (C) det k Sym(j) (k, j det Sym(j) j j j + 1 ρ dominant integral weight (k + j, k) ρ k,j Γ Sp(2, R) Γ\Sp(2, R) H 2 ( ) A B f(γz) = ρ k,j (CZ + D)f(Z) γ = Γ C D γ Γ ρ k,j Γ j = 0 j > 0 Γ\H 2 S k,j (Γ) Γ Sp(2, Z) Sp(2, Q p ) ( Sp(2, Q) Sp(2, Z) Z pz Z Z B(p) = Z Z Z Z pz pz Z Z Sp(2, Z), pz pz pz Z

5 Z Z Z Z Γ 0 (p) = Z Z Z Z pz pz Z Z Sp(2, Z), pz pz Z Z Z pz Z Z Γ 0(p) = Z Z Z Z Z pz Z Z Sp(2, Z) pz pz pz Z Z pz Z Z Γ 0(p) = Z Z Z p 1 Z pz pz Z Z Sp(2, Q) pz pz pz Z Z pz Z Z K(p) = Z Z Z p 1 Z Z pz Z Z Sp(2, Q) pz pz pz Z Z Z p 1 Z p 1 Z Sp(2, Z) = Z Z p 1 Z p 1 Z pz pz Z Z Sp(2, Q) pz pz Z Z w = p 0 0 p w K(p) normalize Sp (2, Z) = w 1 Sp(2, Z)w, Γ 0(p) = w 1 Γ 0(p)w S k,j (Γ 0(p)) S k,j (Γ 0(p)) ( ) w 1 0 K(p) 0 p B(p) (minimal parahoric) [4], [2], [6] Γ 0 (p), Γ 0(p), Γ 0(p) Γ 0 (p) Sp(2, Z) SL 2 (Z) Γ (1) 0 (p) = {( ) a b c d } SL 2 (Z); c 0 mod p (1)

6 6 4. Sp(2, Q) 1982 Γ 0(p), Γ 0(p) p = 2, 3 Sp(2, Q) p = 2 Sp(2, F 2 ) S k (Γ(2)) (Γ(2) p = 3 Freitag, Salvati-Manni Γ(3) Sp(2, F 3 )/{±1 4 } dim S k (Γ 0(3)) = dim S k (Γ 0(3)) f 1 > f 2, Sp(2, Q) j > 0 local data G Sp(2, Q) local data local data Γ 0(p), Γ 0(p) p G p GSp(2, Q p ) local data G p local data [1] GSp(2, Q p ) local data Γ 0(p) p = 2, p = 3 p = 2, 3 ( Γ 0 (p) K(p) p = 2, 3 p = 2, 3 Sp(2, Q) k 5 Lefschetz fixed point theorem k 5 k 3 (cf. [5])

7 Dan Petersen k 3 Theorem 1. k 3, j 0 j > 0 p = 2, 3 j > 0 k = 3, 4 dim S k,j (Γ 0(p)) + dim S k,j (Γ 0(p)) dim S k,j (Γ 0 (p)) 2 dim S k,j (K(p)) = dim M k+j 3,k 3 (U(p)) δ k3 δ j0 (dim S new j+2 (Γ (1) 0 (p)) + δ j0 ) (dim S new 2k+j 2(Γ (1) 0 (p)) + dim S 2k+j 2 (SL 2 (Z))). S new (Γ (1) 0 (p)) newforms δ Kronecker dim S k,j (Γ 0(p)) = dim S k,j (Γ 0(p)) 5. Theorem 1 2 dim S k,j (Γ 0(p)) dim S k,j (Γ 0 (p)) 2 dim S k,j (K(p)) dim M k+j 3,k 3 (U(p)) Γ 0(p), Γ 0 (p), K(p) p-adic Roberts and Schmidt [12] p-adic Γ 0(p), Γ 0 (p), K(p) fixed vectors a, b, c c 0 = 2a b 2c 7

8 8 c 0 1, 0, 1, 2 c 0 = 1 Γ 0 (p) K(p) Γ 0(p) Γ 0 (p) Maass p [7], Boecherer- Schulze-Pillot [14], [13] M k+j 3,k 3 (U(p)) M k+j 3,k 3 (U(p)) Atkin-Lehner involution f S k (Γ (1) 0 (p)) f k W p = ( pτ) k f(pτ) W p 1 ±1. S new,+ (Γ (1) 0 (p)), S new, (Γ (1) 0 (p)) new forms W p +1 1 Conjecture 2. (1) k 3 S new 2k 2(Γ (1) 0 (p)) S 2k 2 (SL 2 (Z)) M k 3,k 3 (U(p)). (2) k 3 j 0, S new,± j+2 (Γ (1) 0 (p)) S new, 2k+j 2 (Γ(1) 0 (p)) M k+j 3,k 3 (U(p)). (3) k 3 j 0 S new j+2 (Γ (1) 0 (p)) S 2k+j 2 (SL 2 (Z)) M k+j 3,k 3 (U(p))). j + 2 2k + j 2 (1) 2 2k 2 j + 2 f, 2k + j 2 g F L p L(s, F ) = L(s k + 2, f)l(s, g) L f 2 L(s k + 2) =

9 ζ(s k + 2)ζ(s k + 1) Sp(2, Z) Sp(2, Z) p Roberts and Schmidt [12] fixed vector Sp(2, Z) Γ 0 (p) Γ 0(p) K(p) c 0 = = 1 M k+j 3,k 3 (U(p)) Sp(2, Z) k S 2k 2 (SL 2 (Z)) k odd M k 3,k 3 (U(p)) Gross Ihara algebraic modular forms 6. : c 0 = 0 c 0 = 1 c 0 = 0, 1, 2 c 0 = 0 Sp(2, Q) p = 2, k = 12 3 λ(3) = Γ 0 (2) Γ 0(2) K(2) c 0 = = 0 T (n) (n 2) M 9,9 (U(2)) = 0 9

10 10 middle parahoric new form Roberts and Schmidt 7. c 0 = 2 [11] p = 3 f 1 = f 2 = 8 M 8,8 (U(3)) 6 Euler 2 factor [11] Euler 3 11 Freitag Salvati-Manni dim S 11 (Γ 0 (3)) = 0, dim S 11 (K(3)) = 1, dim S 11 (Γ 0(3)) = dim S 11 (Γ 0(3)) = 2 Γ 0(p) K(p) S 11 (Γ 0(3)) K(3) Euler 2 factor (1 12( )2 s s )(1 12( )2 s s ) Euler 2 factor S 11 (Γ 0(3)) S 11 (Γ 0(3)) w M 8,8 (U(3)) L good prime c 0 = 2 M k+j 3,k 3 (U(p)) L M k+j 3,k 3 (U(p)) c 0 = 2 Γ 0 (p)

11 bad prime p Γ 0(p) bad prime G p 1 c 0 = c 0 = 1 Roberts and Schmidt GSp(2, Q p ) Schmidt Schmidt c 0 Roberts and Schmidt [8]

12 12 [9] References [1] K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion harmitian forms. J.Fac.Sci.Univ.Tokyo Sect.IA Math. 27(1980) ; (II) 28 (1982), ; (III) 30 (1983), [2] K. Hashimoto and T. Ibukiyama. On relations of dimensions of automorphic forms of Sp(2, R) and its compact twist Sp(2) (II). Advanced Studies in Pure Math. 7 (1985), [3] T. Ibukiyama. On symplectic Euler factors of genus two. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), [4] T. Ibukiyama. On relations of dimensions of automorphic forms of Sp(2, R) and its compact twist Sp(2) (I). Advanced Studies in Pure Math. 7 (1985), [5] T. Ibukiyama, Dimension formulas of Siegel modular forms of weight 3 and supersingular abelian surfaces, in Proceedings of the 4-th Spring Conference, Abelian Varieties and Siegel Modular Forms (2007), [6] T. Ibukiyama, Paramodular forms and compact twist, Automorphic Forms on GSp(4), Proceedings of the 9-th Autumn Workshop on Number Theory, Ed. M. Furusawa (2007), [7] T. Ibukiyama, Saito Kurokawa lifting of level N and practical construction of Jacobi forms, Kyoto J. Math. 52, No. 1 (2012), [8] T. Ibukiyama, Conjectures on correspondence of symplectic modular forms of middle parahoric type and Ihara lifts, Res. Math. Sci. 5 (2018), no. 2, Paper No. 18, 36 pp, Springer Verlag. [9] x+467 pp. [10] T. Ibukiyama and Y. Ihara. On automorphic forms on the unitary symplectic group Sp(n) and SL 2 (R), Math. Ann. 278 (1987), [11] Y. Ihara, On certain arithmetical Dirichlet series, J. Math. Soc. Japan 16 (1964), [12] B. Roberts and R. Schmidt, Local new forms for GSp(4), Lecture Notes in Mathematics, 1918, Springer Verlag, Berlin (2007). [13] R. Schmidt, Iwahori-spherical representations of GSp(4) and Siegel modular forms of degree 2 with square-free level. J. Math. Soc. Japan 57 (2005), [14] R. Schmidt, On classical Saito-Kurokawa liftings, J. reine angew. Math. 604(2007), Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, Japan address: ibukiyam@math.sci.osaka-u.ac.jp

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

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,