Modular forms on the compact symplectic groups Tomoyoshi Ibukiyama, - - PowerPoint PPT Presentation

Modular forms on the compact symplectic groups

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Contents

1

Aim of my talk

2

Symplectic groups of degree two

3

Automorphic forms on compact twist

4

Examples and dimensions

5

Supersingular geometry

6

Comparison with Siegel modular forms

7

Ihara lifts and conjectures

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Aim of my talk

Arithmetic of quaternion hermitian forms and geometry of supersingular abelian varieties. Dimension formulas of such modular forms and comparison with those of Siegel modular forms. Precise conjecture on images of Ihara lifts which are compact version of Saito-Kurokawa lifts.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Symplectic groups over real number fields

We define Sp(n, R) = {g ∈ SL2n(R); gJ tg = J} where J = (0n −1n 1n 0n ) . (split symplectic group → Siegel modular) There are other groups over R which are isomorphic to Sp(n, C) over the scalar extension to C. One of which is described as follows Let H = R + Ri + Rj + Rk (i2 = j2 = −1, k = ij = −ji) be the unique division quaternion algebra over R. Define compact (unitary) symplectic group as follows. USp(n) = {g ∈ Mn(H) : gg ∗ = 1n}. Here for g = (gij), we write g ∗ = tg = (gji) Since H ⊗R C ∼ = M2(C), we have USp(n) ⊗ C ∼ = Sp(n, C).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Symplectic group over Q

We define algebraic groups over Q isomorphic to Sp(n, C) over C. Sp(n, Q) = Sp(n, R) ∩ M4(Q).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Symplectic group over Q

We define algebraic groups over Q isomorphic to Sp(n, C) over C. Sp(n, Q) = Sp(n, R) ∩ M4(Q). We define a group Gn by Gn = Gn(Q) = {g ∈ Mn(D); gg ∗ = n(g)1n for some n(g) ∈ Q×}. Here let D = Q + Qα + Qβ + Qαβ (α2 < 0, β2 < 0, αβ = −βα) be a division quaternion algebra over Q with D ⊗Q R = H. We denote by G 1

n the subgroup such that

n(g) = 1. Then we have G 1

n (R) ∼

= USp(n). (compact twist) Here g ∗ = tg = (gji) and ¯ is the main involution: x + yα + zβ + wαβ = x − yα − zβ − wαβ.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Symplectic group over Q

We define algebraic groups over Q isomorphic to Sp(n, C) over C. Sp(n, Q) = Sp(n, R) ∩ M4(Q). We define a group Gn by Gn = Gn(Q) = {g ∈ Mn(D); gg ∗ = n(g)1n for some n(g) ∈ Q×}. Here let D = Q + Qα + Qβ + Qαβ (α2 < 0, β2 < 0, αβ = −βα) be a division quaternion algebra over Q with D ⊗Q R = H. We denote by G 1

n the subgroup such that

n(g) = 1. Then we have G 1

n (R) ∼

= USp(n). (compact twist) Here g ∗ = tg = (gji) and ¯ is the main involution: x + yα + zβ + wαβ = x − yα − zβ − wαβ. For a prime q, we put Dq = D ⊗Q Qq for q-adic number field Qq and put Gn(Qq) = {g ∈ Mn(Dq); gg ∗ = n(g)1n for some n(g) ∈ Q×

q }.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Arithmetic of Gn and quaternion hermitian forms

We fix D and a maximal order O of D. Left O lattices: A lattice L in Dn (regarded as a 4n dimensional vector space over Q) is called a left O-lattice if OL ⊂ L.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Arithmetic of Gn and quaternion hermitian forms

We fix D and a maximal order O of D. Left O lattices: A lattice L in Dn (regarded as a 4n dimensional vector space over Q) is called a left O-lattice if OL ⊂ L. Class: Two left O-lattices L1, L2 in Dn are said to be in the same class if L1 = L2g for some g ∈ Gn.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Arithmetic of Gn and quaternion hermitian forms

We fix D and a maximal order O of D. Left O lattices: A lattice L in Dn (regarded as a 4n dimensional vector space over Q) is called a left O-lattice if OL ⊂ L. Class: Two left O-lattices L1, L2 in Dn are said to be in the same class if L1 = L2g for some g ∈ Gn. Genus: Two left O-lattices L1, L2 in Dn are said to be in the same genus if L1 ⊗Z Zq = (L2 ⊗Z Zq)gq for some gq ∈ Gn(Qq) for all primes q, where Zq is the ring of q-adic integers.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Arithmetic of Gn and quaternion hermitian forms

We fix D and a maximal order O of D. Left O lattices: A lattice L in Dn (regarded as a 4n dimensional vector space over Q) is called a left O-lattice if OL ⊂ L. Class: Two left O-lattices L1, L2 in Dn are said to be in the same class if L1 = L2g for some g ∈ Gn. Genus: Two left O-lattices L1, L2 in Dn are said to be in the same genus if L1 ⊗Z Zq = (L2 ⊗Z Zq)gq for some gq ∈ Gn(Qq) for all primes q, where Zq is the ring of q-adic integers. Principal genus: The genus containing On is the principal genus.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Arithmetic of Gn and quaternion hermitian forms

We fix D and a maximal order O of D. Left O lattices: A lattice L in Dn (regarded as a 4n dimensional vector space over Q) is called a left O-lattice if OL ⊂ L. Class: Two left O-lattices L1, L2 in Dn are said to be in the same class if L1 = L2g for some g ∈ Gn. Genus: Two left O-lattices L1, L2 in Dn are said to be in the same genus if L1 ⊗Z Zq = (L2 ⊗Z Zq)gq for some gq ∈ Gn(Qq) for all primes q, where Zq is the ring of q-adic integers. Principal genus: The genus containing On is the principal genus. Maximal lattices: A left O-lattice L is called maximal if L is maximal among those lattices M such that two sided O-ideals generated by xy ∗ = ∑n

i=1 xiyi for x, y ∈ M are the same.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Arithmetic of Gn and quaternion hermitian forms

We fix D and a maximal order O of D. Left O lattices: A lattice L in Dn (regarded as a 4n dimensional vector space over Q) is called a left O-lattice if OL ⊂ L. Class: Two left O-lattices L1, L2 in Dn are said to be in the same class if L1 = L2g for some g ∈ Gn. Genus: Two left O-lattices L1, L2 in Dn are said to be in the same genus if L1 ⊗Z Zq = (L2 ⊗Z Zq)gq for some gq ∈ Gn(Qq) for all primes q, where Zq is the ring of q-adic integers. Principal genus: The genus containing On is the principal genus. Maximal lattices: A left O-lattice L is called maximal if L is maximal among those lattices M such that two sided O-ideals generated by xy ∗ = ∑n

i=1 xiyi for x, y ∈ M are the same.

Non-principal genus: If the discriminant of D is a prime p and n ≥ 2, then there are only two genera of maximal lattices, the principal genus Lpr and the non-principal genus Lnpr.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Automorphic forms on “compact twist” Gn

We omit adelic formulation here. We fix a genus L of left O lattices. The number of classes in L is finite and called a class number.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Automorphic forms on “compact twist” Gn

We omit adelic formulation here. We fix a genus L of left O lattices. The number of classes in L is finite and called a class number. We fix representatives L1, . . . , Lh of classes in L. We define an automorphism group of Li by Γi = {γ ∈ Gn; Liγ = Li} ⊂ USp(n). These are finite groups but play a role of discrete subgroups.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Automorphic forms on “compact twist” Gn

We omit adelic formulation here. We fix a genus L of left O lattices. The number of classes in L is finite and called a class number. We fix representatives L1, . . . , Lh of classes in L. We define an automorphism group of Li by Γi = {γ ∈ Gn; Liγ = Li} ⊂ USp(n). These are finite groups but play a role of discrete subgroups. We fix an irreducible representation (τ, V ) of USp(n) over C, which plays a role of weight of automorphic forms.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Automorphic forms on “compact twist” Gn

We omit adelic formulation here. We fix a genus L of left O lattices. The number of classes in L is finite and called a class number. We fix representatives L1, . . . , Lh of classes in L. We define an automorphism group of Li by Γi = {γ ∈ Gn; Liγ = Li} ⊂ USp(n). These are finite groups but play a role of discrete subgroups. We fix an irreducible representation (τ, V ) of USp(n) over C, which plays a role of weight of automorphic forms. We put V Γi = {v ∈ V : τ(γi)v = v for all γ ∈ Γi}. Then the space of automorphic forms of weight τ of L is defined to be Mτ(L) = ⊕h

i=1V Γi.

When τ is trivial, the class number h = dim Mτ(L).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Harmonic polynomials on H2

Identify H2 ∼ = R8 by mapping (x, y) to (x1, x2, x3, x4, y1, y2, y3, y4) for x = x1 + x2i + x3j + x4k, y = y1 + y2i + y3j + y4k. We put ∆x,y =

4

∑

i=1

∂2 ∂x2

i

+

4

∑

i=1

∂2 ∂y 2

i

. For λ = λ1 + λ2i + λ3j + λ4k ∈ H, we put n(λ) = λ2

1 + λ2 2 + λ2 3 + λ2 4,

∆λ =

4

∑

i=1

∂2 ∂λ2

i

. By Harmℓ(H2) the space of homogeneous harmonic polynomials f (x, y) of even degree ℓ such that ∆x,yf = 0.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

• Irred. decomposition of harmonic polynomials.

For integers a, b with a ≥ b ≥ 0 and a + b = l, we put Va,b = { f ∈ Harmℓ(H2); f (λx, λy) = n(λ)bφ(x, y, λ) ∆λφ = 0 } . Let Syma−b be the symmetric tensor representation of SU(2) ∼ = USp(1). Let τa,b be the irreducible representatin of USp(2) corresponding to the Young diagram (a, b) (a ≥ b ≥ 0). For even integer l ≥ 0, we have Harmℓ(H2) = ⊕a≥b≥0,a+b=lVa,b. The space Va,b is the representation space of Syma−b ⊗ τa,b of USp(1) × USp(2) by the action f (x, y) → f (u(x, y)g) for u ∈ USp(1) ∼ = SU(2) and g ∈ USp(2). In particular, if a = b, then Va,a is an irreducible representation space of USp(2) and ϕ(x, y, λ) = f (x, y).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Example of automorphic forms

dim M12,12(Lpr(2)) = 6. We write H ∋ w = w1 + w2i + w3j + w4k, and n(w) = w 2

1 + w 2 2 + w 2 3 + w 2 4 . For x, y ∈ H, we put z = yx. For

any positive integer i, we put ti = zi

1 + zi 2 + zi 3 + zi

• 4. Then a basis of

M12,12(Lpr(2)) is given by

P12a = (z2

1 − z2 2 )(z2 1 − z2 3 )(z2 1 − z2 4 )(z2 2 − z2 3 )(z2 2 − z2 4 )(z2 3 − z2 4 ),

P12b = 226512n(x)6n(y)6 − 169884n(x)5n(y)5(n(x)2 + n(y)2) + 70785n(x)4n(y)4(n(x)4 + n(y)4) − 15730n(x)3n(y)3(n(x)6 + n(y)6) + 1716n(x)2n(y)2(n(x)8 + n(y)8) − 78n(x)n(y)(n(x)10 + n(y)10) + n(x)12 + n(y)12, P12c = 279(n(x)12 + n(y)12)/169380640 − 837n(x)n(y)(n(x)10 + n(y)10)/6514640 + 837n(x)2n(y)2(n(x)10 + n(y)10)/296120 − 1915n(x)3n(y)3(n(x)6 + n(y)6)/139984 + 6963n(x)4n(y)4(n(x)4 + n(y)4)/279968 − 137n(x)5n(y)5(n(x)2 + n(y)2)/5384 − 428n(x)n(y)(n(x)6 + n(y)6)t4/8749 + 2568n(x)2n(y)2(n(x)4 + n(y)4)t4/8749 − 428/673n(x)3n(y)3(n(x)2 + n(y)2)t4 + n(x)4n(y)4t4 + 1070(n(x)4 + n(y)4)t2

4 /8749

− 428/673n(x)n(y)(n(x)2 + n(y)2)t2

4 + 330/673nx2ny2t2 4 + 476t3 4 /2019

+ 1712(n(x)6 + n(y)6)t6/43745 − 5992n(x)n(y)(n(x)4 + n(y)4)t6/43745 + 448/673n(x)n(y)t4t6 − 12840(n(x)4 + n(y)4)t8/61243 + 5136n(x)n(y)(n(x)2 + n(y)2)t8)/4711 − 7656n(x)2n(y)2t8/4711 − 336t4t8/673, Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

P12d = 181(n(x)12 + n(y)12)/215575360 − 543n(x)n(y)(n(x)10 + n(y)10)/8291360 + 543n(x)2n(y)2(n(x)8 + n(y)8)/376880 − 13661n(x)3n(y)3(n(x)6 + n(y)6)/1959776 − 35181n(x)4n(y)4(n(x)4 + n(y)4)/3919552 + 3753n(x)5n(y)5(n(x)2 + n(y)2)/37688 − 873n(x)n(y)(n(x)6 + n(y)6)t4/34996 + 20571n(x)2n(y)2(n(x)4 + n(y)4)t4/69992 − 723/673n(x)3n(y)3(n(x)2 + n(y)2)t4 − 2365(n(x)4 + n(y)4)t2

4 /69992

+ 473n(x)n(y)(n(x)2 + n(y)2)t2

4 /2692 + 4851n(x)2n(y)2t2 4 /1346

+ 473n(x)n(y)(n(x)2 + n(y)2)t2

4 /2692 − 1122t3 4 /673 + 873(n(x)6 + n(y)6)t6/43745

− (11468n(x)n(y)(n(x)4 + n(y)4)t6/43745 + n(x)2n(y)2(n(x)2 + n(y)2)t6+ − 3168/673n(x)n(y)t4t6 + 7095(n(x)4 + n(y)4)t8/122486 − 1419n(x)n(y)(n(x)2 + n(y)2)t8/4711 − 2970n(x)2n(y)2t8/4711 + 2376t4t8/673, P12e = 1943(n(x)12 + n(y)12)/1552142592 − 1943n(x)n(y)(n(x)10 + n(y)10)/19899264 + 1943n(x)2n(y)2)(n(x)8 + n(y)8)/904512 − 473225n(x)3n(y)3(n(x)6 + n(y)6))/70551936 − 137455n(x)4n(y)4(n(x)4 + n(y)4)/15678208 + 17305n(x)5n(y)5(n(x)2 + n(y)2)/301504 − 3635n(x)n(y)(n(x)6 + n(y)6)t4/69992 + 10905n(x)2n(y)2(n(x)4 + n(y)4)t4/34996 − 3635n(x)3n(y)3(n(x)2 + n(y)2)t4/5384 + 18175(n(x)4 + n(y)4)t2

4 /139984

− 3635n(x)n(y)(n(x)2 + n(y)2)t2

4 /5384 + 8265n(x)2n(y)2t2 4 /2692 − 1700t3 4 /2019

+ 727(n(x)6 + n(y)6)t6/17498 − 5089n(x)n(y)(n(x)4 + n(y)4)t6/34996 + n(x)3n(y)3t6 − 5089n(x)n(y)(n(x)4 + n(y)4)t6/34996 − 1600/673n(x)n(y)t4t6 − 54525(n(x)4 + n(y)4)t8/244972 + 10905n(x)n(y)(n(x)2 + n(y)2)t8/9422 − 11595n(x)2n(y)2t8/4711 + 1200t4t8/673), Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

P12f = 9557(n(x)12 + n(y)12)/34147137024 − 9557n(x)n(y)(n(x)10 + n(y)10)/437783808 + 9557n(x)2n(y)2(n(x)8 + n(y)8))/19899264 − 204145n(x)3n(y)3(n(x)6 + n(y)6))/141103872 − 73895n(x)4n(y)4(n(x)4 + n(y)4)/31356416 + 8401n(x)5n(y)5(n(x)2 + n(y)2)/603008 − 1655n(x)n(y)(n(x)6 + n(y)6)t4/139984 + 4965n(x)2n(y)2(n(x)4 + n(y)4)t4/69992 − 1655n(x)3n(y)3(n(x)2 + n(y)2)t4/10768 + 8275(n(x)4 + n(y)4)t2

4 /279968

− 1655n(x)n(y)(n(x)2 + n(y)2)t2

4 /10768 + 1215n(x)2n(y)2t2 4 /1346

− 1655n(x)n(y)(n(x)2 + n(y)2)t2

4 /10768 + 8275(n(x)4 + n(y)4)t2 4 /279968

+ 2465t3

4 /8076 + 331(n(x)6 + n(y)6)t6/34996 − 2317n(x)n(y)(n(x)4 + n(y)4)t6/69992

− 2205n(x)n(y)t4t6/1346 + t2

6 − 24825(n(x)4 + n(y)4)t8/489944

+ 4965n(x)n(y)(n(x)2 + n(y)2)t8/18844 + 2175n(x)2n(y)2t8/18844 − 435t4t8/673 Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Dimension formulas for compact twist for n = 2

Class number formulas for Lpr, Lnpr and Umin by Hashimoto and Ibukiyama (1980s) for any division algebra D/Q when n = 2. By trace formula, the calculation of class numbers is applicable directly to the calculation of Mτ(Lpr) and Mτ(Lnpr). We consider elements gi ∈ USp(2) with principal polynomials fi(±x) where f1(x) = (x − 1)4 f2(x) = (x − 1)2(x + 1)2 f3(x) = (x − 1)2(x2 + 1) f4(x) = (x − 1)2(x2 + x + 1) f5(x) = (x − 1)2(x2 − x + 1) f6(x) = (x2 + 1)2 f7(x) = (x − 2 + x + 1) f8(x) = (x2 + 1)(x2 + x + 1) f9(x) = (x2 + x + 1)(x2 − x + 1), f10(x) = x4 + x3 + x2 + x + 1, f11(x) = x4 + 1, f12(x) = x4 − x2 + 1. These are only possibile principal polynomials of elements in Γi.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Example of dimensions of automorphic forms on G2

For an irreducible rep. τ of USp(2), we have dim Mτ(Lpr(p)) =

12

∑

i=1

Tr(τ(gi))Hcpt

i

, where Tr(τ(gi)) is given by H. Weyl and Hcpt

i

are given as follows.

Hcpt

1

= 1 26 · 32 · 5(p − 1)(p2 + 1), Hcpt

2

= 1 26 · 32 (p − 1)2 × { 7 if p ̸= 2, 13 if p = 2, Hcpt

3

= 1 24 · 3(p − 1) ( 1 − (−1 p )) , Hcpt

4

= Hcpt

5

= 1 23 · 32 (p − 1) ( 1 − (−3 p )) , Hcpt

6

= 1 25 ( 1 − (−1 p )) (1 − δp2) + 5 25 · 3(p − 1),

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Hcpt

7

= 1 22 · 32 ( 1 − (−3 p )) + 1 2 · 32 (p − 1)(1 − δp3), Hcpt

8

= 1 22 · 3 ( 1 − (−1 p )) ( 1 − (−3 p )) , Hcpt

9

=      1 32 ( 1 − (−3 p ))2 if p ̸= 2, 5/18 if p = 2, Hcpt

10 = 1

5 ×    1 if p = 5, 4 if p ≡ 4 mod 5,

• therwise,

Hcpt

11 = 1

23 ×        1 if p = 2, if p ≡ 1 mod 8, 2 if p ≡ 3, 5 mod 8, 4 if p ≡ 7 mod 8, Hcpt

12 =

1 22 · 3 ( 1 − (−3 p )) .

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Supersingular abelian Varieties

We fix a prime p. An elliptic curve E is called supersingular if it has no p-tortion (etale) points. Or equivalently, D = End(E) ⊗Z Q is the definite quaternion algebra of discriminant p.

Theorem (Deuring)

(1) The number of isomorphism classes of supersingular E is equal to the class number H of D. (2) E is defined over Fp2. The number of supersingular E which has a model over Fp is equal to 2T − H, where T is the type number of D defined as the number of isomorphism classes of maximal orders of D.

Theorem (Shioda, Ogus, Deligne)

Assume n ≥ 2 and let E, E1, E2, . . . , En be supersingular elliptic

• curves. Then we have E n ∼

= E1 × · · · × En. (uniqueness of superspecial abelian varieties).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Polarizations

We say E n is a superspecial abelian variety if E is a supersingular elliptic curve. There exists only one superspecial abelian variety up to isomorphism. We have nothing to count here.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Polarizations

We say E n is a superspecial abelian variety if E is a supersingular elliptic curve. There exists only one superspecial abelian variety up to isomorphism. We have nothing to count here. We say an abelian variety A is supersingular if A ∼ E n (isogeny). There are infinitely many such A up to isomorphism. We can consider the moduli of all such A with polarization. The moduli is not irreducible and we will count the number of components.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Polarizations

We say E n is a superspecial abelian variety if E is a supersingular elliptic curve. There exists only one superspecial abelian variety up to isomorphism. We have nothing to count here. We say an abelian variety A is supersingular if A ∼ E n (isogeny). There are infinitely many such A up to isomorphism. We can consider the moduli of all such A with polarization. The moduli is not irreducible and we will count the number of components. Consider a divisor X on the superspecial E n defined by X = ∑n

i=1 E i−1 × {0} × E n−i. We define an isogeny ϕX by

ϕX(t) = Cl(Xt − X) : E n ∋ t → ϕX(t) ∈ E n = Pic0(E n), where Cl is the linear equivalent class and Xt is the translation

• f X by t. This is an isomorphism. An isogeny λ : E n → E n is

called polarization if h = ϕ−1

X λ ∈ Mn(O) ∼

= Mn(End(E)) satisfies h∗ = h and h > 0. We can talk on the genus to which λ belongs.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Class numbers and geometry

The polarization is called principal if it is an isomorphism. The polarization is called ”non-principal” if ϕ−1

X λ belongs to Lnpr.

Theorem (Ibu. Katsura and Oort)

The number of principal polarizations on E n is equal to the class number h(Lpr) of the principal genus Lpr. Denote by Sn,p the locus of supersingular abelian varieties in the moduli of principally polarized abelian varieties of dimension n.

Theorem (Li and Oort, Katsura and Oort)

The number of irreducible components of Sn,p is equal to h(Lpr) if n is odd and to h(Lnpr) if n is even.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

G-type number and geometry

We assume that n ≥ 2. Let L be a genus of quaternion hermitian

• lattices. Let L1, . . . , Lh be representatives of classes in L. We put

Ri = {g ∈ Mn(B); Lig ⊂ Li} Then all Ri are maximal orders and conjugate by GLn(B) for n ≥ 2, so not interesting unless n = 1, which is the Deuring case. Change of the definition: the number of G-conjugacy classes in {R1, . . . , Rh} is called a G-type number T(L) of L.

Theorem (preprint 2016)

(1) The number of principal polarizations of E n which have a model

• ver Fp is 2T(Lpr) − H(Lpr).

(2) Sn,p is defined over Fp2. The number of irrecudible component of Sn,p which has a model over Fp is equal to 2T(Lpr) − H(Lpr) if n is

• dd and to 2T(Lnpr) − H(Lnpr) if n is even.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Example of type numbers

Theorem (preprint 2016)

The number 2T(Lnpr) − H(Lnpr) is is given explicitly as follows. When p = 2, 3, 5, it is one. If p ≥ 7, then for p ≡ 1 mod 4, we have 1 25 · 3 ( 9 − 2 (2 p )) B2,χ + 1 24h(√−p) + 1 23h( √ −2p) + 1 22 · 3 ( 3 + (2 p )) h( √ −3p), and for p ≡ 3 mod 4, we have 1 25 · 3B2,χ + 1 24 ( 1 − (2 p )) h(√−p) + 1 23h( √ −2p) + 1 22 · 3h( √ −3p).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Eichler’s classic theory

Theorem (Eichler)

For any integer k ≥ 2, CE2δk2 ⊕ Snew

k

(Γ(1)

0 (p)) ∼

= Mk−2(Lpr). Here k − 2 indicates the symmetric tensor representation of order k − 2 of USp(1) ∼ = SU(2) and Γ(1)

0 (p) = {( a b c d ) ; c ≡ 0 mod p}.

Question: How can we generalize this neat isomorphism to Siegel modular forms of degree two? (raised by Ihara in 1963. Of course now this is a part of Langlands conjecture.)

：

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Eichler’s classic theory

Theorem (Eichler)

For any integer k ≥ 2, CE2δk2 ⊕ Snew

k

(Γ(1)

0 (p)) ∼

= Mk−2(Lpr). Here k − 2 indicates the symmetric tensor representation of order k − 2 of USp(1) ∼ = SU(2) and Γ(1)

0 (p) = {( a b c d ) ; c ≡ 0 mod p}.

Question: How can we generalize this neat isomorphism to Siegel modular forms of degree two? (raised by Ihara in 1963. Of course now this is a part of Langlands conjecture.) More precisely, what are discrete subgroups, weights and new forms(i.e. local automorphic representations)?

：

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Eichler’s classic theory

Theorem (Eichler)

For any integer k ≥ 2, CE2δk2 ⊕ Snew

k

(Γ(1)

0 (p)) ∼

= Mk−2(Lpr). Here k − 2 indicates the symmetric tensor representation of order k − 2 of USp(1) ∼ = SU(2) and Γ(1)

0 (p) = {( a b c d ) ; c ≡ 0 mod p}.

Question: How can we generalize this neat isomorphism to Siegel modular forms of degree two? (raised by Ihara in 1963. Of course now this is a part of Langlands conjecture.) More precisely, what are discrete subgroups, weights and new forms(i.e. local automorphic representations)? One Answer for n = 2：Instead of Γ(1)

0 (p), take various parahoric

• subgroups. There are many parahoric subgroups, so describe

conjectures for each case.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Parahoric subgroups of Sp(2, R) for a prime p

We define the left groups by “(the right set) ∩Sp(2, Q)”. Sp(2, Z) : M4(Z). B(p) :     Z pZ Z Z Z Z Z Z pZ pZ Z Z pZ pZ pZ Z     Γ0(p) :     Z Z Z Z Z Z Z Z pZ pZ Z Z pZ pZ Z Z     Γ′

0(p) :

    Z pZ Z Z Z Z Z Z Z pZ Z Z pZ pZ pZ Z     Γ′′

0(p) :

    Z pZ Z Z Z Z Z p−1Z pZ pZ Z Z pZ pZ pZ Z     K(p) :     Z pZ Z Z Z Z Z p−1Z Z pZ Z Z pZ pZ pZ Z     Sp(2, Z)∗ :     Z Z p−1Z p−1Z Z Z p−1Z p−1Z pZ pZ Z Z pZ pZ Z Z    

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Relations for k ≥ 3, even j ≥ 0 (some exception)

Theorem

dim Sk,j(B(p)) − dim Sk,j(Γ0(p)) − dim Sk,j(Γ′

0(p)) − dim Sk,j(Γ′′ 0(p)) +

dim Sk,j(K(p)) + dim Sk,j(Sp(2, Z)) + dim Sk,j(Sp∗(2, Z)) = dim Mk+j−3,k−3(Umin)−dim Mk+j−3,k−3(Lpr)−dim Mk+j−3,k−3(Lnpr).

Theorem

dim Sk,j(K(p)) − 2 dim Sk,j(Sp(2, Z)) + δk3δj0 = dim Mk−3,j(Lnp) − (dim Snew

j+2 (Γ(1) 0 (p)) + δj0) × dim S2k+j−2(SL2(Z)).

Theorem

dim Sk,j(Γ′

0(p))+dim Sk,j(Γ′′ 0(p))−dim Sk,j(Γ0(p))−2 dim Sk,j(K(p)) =

dim Mk+j−3,k−3(Lpr) − δk3δj0 − (dim Snew

j+2 (Γ(1) 0 (p)) + δj2) ×

(dim Snew

2k+j−2(Γ(1) 0 (p)) + dim S2k+j−2(SL2(Z))).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Various theory of liftings to degree two

Saito-Kurokawa lift (1970s) (also there is a general level version) S2k−2(SL2(Z)) → Sk(Sp(n, Z)).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Various theory of liftings to degree two

Saito-Kurokawa lift (1970s) (also there is a general level version) S2k−2(SL2(Z)) → Sk(Sp(n, Z)). Yoshida lift (1980s) Sj+2(Γ(1)

0 (p)) × S2k+j−2(Γ(1) 0 (p)) → Sk,j(Γ(2) 0 (p)).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Various theory of liftings to degree two

Saito-Kurokawa lift (1970s) (also there is a general level version) S2k−2(SL2(Z)) → Sk(Sp(n, Z)). Yoshida lift (1980s) Sj+2(Γ(1)

0 (p)) × S2k+j−2(Γ(1) 0 (p)) → Sk,j(Γ(2) 0 (p)).

Ihara lift (1963). This contains both liftings of Saito-Kurokawa type and Yoshida type, but the target is automorphic forms of the compact twist. These are compatibly considered in the conjectures on correspondence.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Automorphic forms on G1 × G2

We recall that for even integer l ≥ 0, we have Harml(H2) = ⊕a≥b≥0,a+b=lVa,b. We denote by J1,. . . ,Jh0 representatives of left O-ideal classes in D. Let Oi be the right order of Ji and Ei = O×

i . We write L1, . . . , Lh

the representatives of classes in a genus L of D2 and Γκ the automorphism groups. We put W =

h0

∑

i=1 h

∑

κ=1

V Ei×Γκ

a,b

, where V Ei×Γκ

a,b

is the Ei × Γκ invariant vectors in Va,b. This space W is obviously isomophic to a product of automorphic forms on G1 and those on G2. So we may write F = (F1, F2).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

A theory of Ihara lift: theta functions

A theory of Ihara lift was initiated by Ihara’s master thesis in 1963, partly published in 1964, and augmented in Ibukiyama-Ihara in 1987. For a Hecke eigenform F = (F1, F2) = (Fiκ), define ϑF,iκ by ϑF,iκ(τ) =

∞

∑

m=0

∑

(x,y)∈ILıκ, niκ(x)+niκ(y)=m

e2πimτ (τ ∈ H1). Then ϑF = ∑h0

i=1

∑h

κ=1 |Ei|−1|Γi|−1ϑF,iκ(τ) ∈ Aa+b+4(Γ0(p) for Lpr.

Theorem

For Lpr, if dϑF (τ)

dq

̸= 0 for q = e2πiτ and ϑF is an eigenform, then L(s, F2) = L(s − b + 2, F1)L(s, ϑF). This is a compact analogue of Saito-Kurokawa lifts (1970s) when F1 is constant, and of Yoshida lifts(1980s) when F1 is not constant.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

New conjectures on Ihara lifts

For f ∈ Sk(Γ(1)

0 (p)), we write f |kWp = (√pτ)−kf (pτ). Eigenvalues

• f Wp is ±1. We consider the case L = Lpr.

Conjecture

(1) If k ≥ 3 is odd and j = 0, then we have an injection Snew

2k−2(Γ0(p)) + S2k−2(SL2(Z)) → Mk+j−3,j−3(Upr).

(2) For any k ≥ 3 and even j ≥ 0, we have an injection Snew,±

j+2

(Γ(1)

0 (p)) × Snew,∓ 2k+j−2(Γ(1) 0 (p)) → Mk+j−3,k−3(Upr(p)).

(3) For any k ≥ 3 and even j ≥ 0, we have an injection Snew

j+2 (Γ(1) 0 (p)) × S2k+j−2(SL2(Z)) → Mk+j−3,k−3(Upr)).

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28

Thank you for your attension.

Tomoyoshi Ibukiyama, (伊吹山知義） Professor Emeritus of Osaka University (大阪大学名誉教授） Modular forms on the compact symplectic groups 16:00-17:00, September 28, 2016 at 上海交通大学数学科学学院 / 28