On Siegel modular forms part II

Abstract

In this paper we compute dimension formulas for rings of Siegel modular forms of genus g = 3. Let denote the main congruence subgroup of level two, the Hecke subgroup of level two and the full modular group. We give the dimension formulas for genus g = 3 for the above mentioned groups and determine the graded ring of modular forms with respect to .

Congruences for Hecke eigenvalues of Siegel modular forms

“…We note that Q(χ It seems difficult to construct this G concretely since dim S 3 20 = 6 by Tsuyumine [45] and Runge [39].…”

A theta relation in genus 4

“…From this point of view the determination of the ring of Siegel modular forms depends on the determination of the kernel of R g → A g . Along these lines the cases g ≤ 3 have been treated successfully (reproducing and enlarging known results) [Ru1], [Ru2]. This paper is an approach to attack the case g = 4.…”

“…It is well-known [Du], [Gl], [Ru1], that a H g -invariant polynomial P ∈ R g is a linear combination of code polynomials of self-dual doubly-even codes if and only if its degree is divisible by 8. In our case (g = 4 and k = 12) we have to consider codes in F 24 2 .…”

“…We get a relation in genus three. In this case there is a defining relation of degree 16 [Ru1]. It is the code polynomial…”

Modular invariance property of association schemes, type II codes over finite rings and finite abelian groups and reminiscences of François Jaeger (a survey)

“…Ozeki [42] or Runge [43]) that if we take the index 2 subgroup H (of order 96) of G defined by The important implication of this fact is that we can understand the space of modular forms completely through the invariant ring of the finite group H. Interestingly enough, this situation can be generalized in several directions. We list some of them in the following table.…”