On certain vector valued Siegel modular forms of degree two

Some vector valued Siegel modular forms of genus 2

is a module over the ring of all modular forms with respect to the group Γ2[4, 8]. We are interested in its structure. By Igusa, the ring of modular forms is generated by the ten classical theta constants θ[m]. The module M contains a submodule N which is generated by 45 Cohen-Rankin brackets {θ[m], θ[n]}. We determine defining relations for this submodule and compute its Hilbert function (Theo...

Some vector valued Siegel modular forms of genus 3 Eberhard

Numerical computation of a certain Dirichlet series attached to Siegel modular forms of degree two

The Rankin convolution type Dirichlet series DF,G(s) of Siegel modular forms F and G of degree two, which was introduced by Kohnen and the second author, is computed numerically for various F and G. In particular, we prove that the series DF,G(s), which shares the same functional equation and analytic behavior with the spinor L-functions of eigenforms of the same weight are not linear combinati...

On the graded ring of Siegel modular forms of degree

The aim of this paper is to give the dimension of the space of Siegel modular forms M k (Γ(3)) of degree 2, level 3 and weight k for each k. Our main result is Theorem dim M k (Γ(3)) = 1 2 (6k 3 − 27k 2 + 79k − 78) k ≥ 4. In other words we have the generating function : ∞ k=0 dim M k (Γ(3))t k = 1 + t + t 2 + 6t 3 + 6t 4 + t 5 + t 6 + t 7 (1 − t) 4. About the space of cusp forms, the dimension ...

A Note on the Growth of Nearly Holomorphic Vector-valued Siegel Modular Forms

Let F be a nearly holomorphic vector-valued Siegel modular form of weight ρ with respect to some congruence subgroup of Sp2n(Q). In this note, we prove that the function on Sp2n(R) obtained by lifting F has the moderate growth (or “slowly increasing”) property. This is a consequence of the following bound that we prove: ‖ρ(Y )F (Z)‖ ∏n i=1(μi(Y ) λ1/2 + μi(Y ) −λ1/2) where λ1 ≥ . . . ≥ λn is th...