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 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 / 28
Contents Aim of my talk 1 Symplectic groups of degree two 2 Automorphic forms on compact twist 3 Examples and dimensions 4 Supersingular geometry 5 Comparison with Siegel modular forms 6 Ihara lifts and conjectures 7 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 / 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. 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 / 28
Symplectic groups over real number fields We define Sp ( n , R ) = { g ∈ SL 2 n ( R ); gJ t g = J } ( 0 n ) − 1 n where J = . (split symplectic group → Siegel modular) 1 n 0 n 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 + R i + R j + R k ( i 2 = j 2 = − 1, k = ij = − ji ) be the unique division quaternion algebra over R . Define compact (unitary) symplectic group as follows. USp ( n ) = { g ∈ M n ( H ) : gg ∗ = 1 n } . Here for g = ( g ij ), we write g ∗ = t g = ( g ji ) Since H ⊗ R C ∼ = M 2 ( C ), we have USp ( n ) ⊗ C ∼ = Sp ( n , C ). 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 / 28
Symplectic group over Q We define algebraic groups over Q isomorphic to Sp ( n , C ) over C . Sp ( n , Q ) = Sp ( n , R ) ∩ M 4 ( Q ). 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 / 28
Symplectic group over Q We define algebraic groups over Q isomorphic to Sp ( n , C ) over C . Sp ( n , Q ) = Sp ( n , R ) ∩ M 4 ( Q ). We define a group G n by G n = G n ( Q ) = { g ∈ M n ( D ); gg ∗ = n ( g )1 n for some n ( g ) ∈ Q × } . Here let D = Q + Q α + Q β + Q αβ ( α 2 < 0, β 2 < 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 ( R ) ∼ n ( g ) = 1. Then we have G 1 = USp ( n ). (compact twist) Here g ∗ = t g = ( g ji ) and ¯ is the main involution: x + y α + z β + w αβ = x − y α − z β − w αβ. 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 / 28
Symplectic group over Q We define algebraic groups over Q isomorphic to Sp ( n , C ) over C . Sp ( n , Q ) = Sp ( n , R ) ∩ M 4 ( Q ). We define a group G n by G n = G n ( Q ) = { g ∈ M n ( D ); gg ∗ = n ( g )1 n for some n ( g ) ∈ Q × } . Here let D = Q + Q α + Q β + Q αβ ( α 2 < 0, β 2 < 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 ( R ) ∼ n ( g ) = 1. Then we have G 1 = USp ( n ). (compact twist) Here g ∗ = t g = ( g ji ) and ¯ is the main involution: x + y α + z β + w αβ = x − y α − z β − w αβ. For a prime q , we put D q = D ⊗ Q Q q for q -adic number field Q q and put G n ( Q q ) = { g ∈ M n ( D q ); gg ∗ = n ( g )1 n for some n ( g ) ∈ Q × q } . 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 / 28
Arithmetic of G n and quaternion hermitian forms We fix D and a maximal order O of D . Left O lattices: A lattice L in D n (regarded as a 4 n dimensional vector space over Q ) is called a left O -lattice if O L ⊂ L . 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 / 28
Arithmetic of G n and quaternion hermitian forms We fix D and a maximal order O of D . Left O lattices: A lattice L in D n (regarded as a 4 n dimensional vector space over Q ) is called a left O -lattice if O L ⊂ L . Class: Two left O -lattices L 1 , L 2 in D n are said to be in the same class if L 1 = L 2 g for some g ∈ G n . 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 / 28
Arithmetic of G n and quaternion hermitian forms We fix D and a maximal order O of D . Left O lattices: A lattice L in D n (regarded as a 4 n dimensional vector space over Q ) is called a left O -lattice if O L ⊂ L . Class: Two left O -lattices L 1 , L 2 in D n are said to be in the same class if L 1 = L 2 g for some g ∈ G n . Genus: Two left O -lattices L 1 , L 2 in D n are said to be in the same genus if L 1 ⊗ Z Z q = ( L 2 ⊗ Z Z q ) g q for some g q ∈ G n ( Q q ) for all primes q , where Z q is the ring of q -adic integers. 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 / 28
Arithmetic of G n and quaternion hermitian forms We fix D and a maximal order O of D . Left O lattices: A lattice L in D n (regarded as a 4 n dimensional vector space over Q ) is called a left O -lattice if O L ⊂ L . Class: Two left O -lattices L 1 , L 2 in D n are said to be in the same class if L 1 = L 2 g for some g ∈ G n . Genus: Two left O -lattices L 1 , L 2 in D n are said to be in the same genus if L 1 ⊗ Z Z q = ( L 2 ⊗ Z Z q ) g q for some g q ∈ G n ( Q q ) for all primes q , where Z q is the ring of q -adic integers. Principal genus: The genus containing O n is the principal genus. 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 / 28
Arithmetic of G n and quaternion hermitian forms We fix D and a maximal order O of D . Left O lattices: A lattice L in D n (regarded as a 4 n dimensional vector space over Q ) is called a left O -lattice if O L ⊂ L . Class: Two left O -lattices L 1 , L 2 in D n are said to be in the same class if L 1 = L 2 g for some g ∈ G n . Genus: Two left O -lattices L 1 , L 2 in D n are said to be in the same genus if L 1 ⊗ Z Z q = ( L 2 ⊗ Z Z q ) g q for some g q ∈ G n ( Q q ) for all primes q , where Z q is the ring of q -adic integers. Principal genus: The genus containing O n 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 x i y i for x , y ∈ M are the same. 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 / 28
Arithmetic of G n and quaternion hermitian forms We fix D and a maximal order O of D . Left O lattices: A lattice L in D n (regarded as a 4 n dimensional vector space over Q ) is called a left O -lattice if O L ⊂ L . Class: Two left O -lattices L 1 , L 2 in D n are said to be in the same class if L 1 = L 2 g for some g ∈ G n . Genus: Two left O -lattices L 1 , L 2 in D n are said to be in the same genus if L 1 ⊗ Z Z q = ( L 2 ⊗ Z Z q ) g q for some g q ∈ G n ( Q q ) for all primes q , where Z q is the ring of q -adic integers. Principal genus: The genus containing O n 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 x i y i 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 L pr and the non-principal genus L npr . 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 / 28
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