Structure Theorems for Certain Vector Valued Siegel Modular Forms of Degree Two

Abstract

The aim of this thesis is to develop two structure theorems for vector valued Siegel modular forms for Igusa’s subgroup Γ_2[2,4], the multiplier system induced by the theta constants and the symmetric square of the standard representation Sym^2 : GL(2,C) → Sym^2(C^2). The thesis rests on the well-known fact that every holomorphic tensor on the upper half-space H_2 that is invariant under Γ_2[2,4] is associated to a vector valued modular form. We define a space of meromorphic tensors that are holomorphic outside a divisor and become holomorphic after a pullback along certain covering maps. Afterwards, we show that Γ_2[2,4]-invariant tensors on H_2 correspond to this particular space of tensors on the quotient space Γ_2[2,4] H_2 with poles along the ramification divisor. As an immediate consequence of a theorem by Runge, Γ_2[2,4] H_2 is embedded into the projective space P^3C and its complement is an analytic set of codimension 2. The ramification divisor is given by 10 simple quadrics. Therefore, we shall describe the aforementioned modular forms with weights in 3Z or 1 + 3Z as rational tensors. The tensors with weights in 6Z or 1 + 6Z possess easily handleable poles along the 10 quadrics. It can be shown that these modular forms are contained in the module generated by the Rankin-Cohen brackets of the four theta series of the second kind. We extend this result to arbitrary weights by a simple argument from algebraic geometry.