Tomoyoshi Ibukiyama

SIMPLE GRADED RINGS OF SIEGEL MODULAR FORMS, DIFFERENTIAL OPERATORS AND BORCHERDS PRODUCTS

In this paper, we show that the graded ring of Siegel modular forms of Γ0(N) ⊂ Sp(2,ℤ) has a very simple unified structure for N = 1, 2, 3, 4, taking Neben-type case (the case with character) for N = 3 and 4. All are generated by 5 generators, and all the fifth generators are obtained by using the other four by means of differential operators, and it is also obtained as Borcherds products. As an appendix, examples of Euler factors of L-functions of Siegel modular forms of Sp(2,ℤ) of odd weight are given.

RANKIN-COHEN TYPE DIFFERENTIAL OPERATORS FOR SIEGEL MODULAR FORMS

Let ℍn be the Siegel upper half space and let F and G be automorphic forms on ℍn of weights k and l, respectively. We give explicit examples of differential operators D acting on functions on ℍn × ℍn such that the restriction of to Z = Z1 = Z2 is again an automorphic form of weight k + l + v on ℍn. Since the elliptic case, i.e. n = 1, has already been studied some time ago by R. Rankin and H. Cohen we call such differential operators Rankin–Cohen type operators. We also discuss a generalisation of Rankin–Cohen type operators to vector valued differential operators.

An explicit formula for Koecher–Maaß Dirichlet series for Eisenstein series of Klingen type

Abstract We give a reasonable expression of the Koecher–Maas Dirichlet series for the Klingen–Eisenstein lift of an elliptic cusp form.

On Automorphic Forms on the Unitary Symplectic Group Sp (n) and SL2 (R).

Soit Sp(n) la forme reelle compacte du groupe symplectique de degre n. On etudie une correspondance entre des formes automorphes, F sur Sp(1)×Sp(n) et ∂ F sur Sp(1,R). On donne des relations explicites entre leurs L-fonctions pour le cas n≥2, r=1

A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences

For modular forms of one variable there is the famous correspondence of Shimura between modular forms of integral weight and half integral weight (cf. [15]). In this paper, we propose a similar conjecture for vector valued Siegel modular forms of degree two and provide numerical evidence and conjectural dimensional equality (Section 1, Main Conjecture 1.1; a short announcement was made in [11].) We also propose a half-integral version of Harder’s conjecture in [4]. Our version is deduced in a natural way from our Main Conjecture. While the original conjecture deals with congruences between eigenvalues of Siegel modular forms and modular forms of one variable our version is stated as a congruence between L-functions of a Siegel cusp form and a Klingen type Eisenstein series. We give here a rough indication of the content of our Main Conjecture. This is restricted to the case of level one, but stated as a precise bijective correspondence as follows. Conjecture For any natural number k ≥ 3 and any even integer j ≥ 0, there is a linear isomorphism

On zeta functions associated to symmetric matrices III : An explicit form of L-functions

In [I-S2], we gave an explicit form of zeta functions associated to the space of symmetric matrices. In this paper, the case of L-functions is treated. In the case of definite symmetric matrices, we show the ratinality of special values of these L-functions.

On zeta functions associated to symmetric matrices, II: Functional equations and special values

New simple functional equations of zeta functions of the prehomogeneous vector spaces consisting of symmetric matrices are obtained, using explicit forms of zeta functions in the previous paper, Part I, and real analytic Eisenstein series of half-integral weight. When the matrix size is 2, our functional equations are identical with the ones by Shintani, but we give here an alternative proof. The special values of the zeta functions at nonpositive integers and the residues are also explicitly obtained. These special values, written by products of Bernoulli numbers, are used to give the contribution of “central” unipotent elements in the dimension formula of Siegel cusp forms of any degree. These results lead us to a conjecture on explicit values of dimensions of Siegel cusp forms of any torsion-free principal congruence subgroups of the symplectic groups of general degree.

On Automorphism Groups of Positive Definite Binary Quaternion Hermitian Lattices and New Mass Formula

Publisher Summary This chapter discusses automorphism groups of positive definite binary quaternion Hermitian lattices and new mass formula. It presents a general method on the way to calculate the multiplicity of a given finite group that appears as the automorphism groups of the lattices, up to isometry, in a fixed genus in a positive definite metric space. The method is then applied to the binary quaternion hermitian cases, motivated by the theory of supersingular abelian varieties developed in Katsura–Oort. The chapter presents a problem that can be solved in principle by the means of the trace formula. It reviews lattices in hermitian spaces. It also presents the classification of G -conjugacy classes of some elements of G 2 and presents the calculation of some local data.

Eisenstein Congruences for SO(4, 3), SO(4, 4), Spinor, and Triple Product L-values

ABSTRACT We work out instances of a general conjecture on congruences between Hecke eigenvalues of induced and cuspidal automorphic representations of a reductive group, modulo divisors of certain critical L-values, in the case that the group is a split orthogonal group. We provide some numerical evidence in the case that the group is SO(4, 3) and the L-function is the spinor L-function of a genus 2, vector-valued, Siegel cusp form. We also consider the case that the group is SO(4, 4) and the L-function is a triple product L-function.

Generalized Bernoulli Numbers

In this chapter we introduce generalized Bernoulli numbers and Bernoulli polynomials. Generalized Bernoulli numbers are Bernoulli numbers twisted by a Dirichlet character, which we define at the beginning of the first section. Bernoulli polynomials are generalizations of Bernoulli numbers with an indeterminate. These two generalizations are related, and they will appear in various places in the following chapters.