Basic vector valued Siegel modular forms of genus two

Abstract

We consider the space $[\Gamma, rr_{0}, v^{r}, \varrho]$ of all vector-valued holomorphic modular forms $f\colon \mathbb{H}_{n} \to \mathcal{Z}$ of transformation type \begin{equation*} f(MZ)=v^{r}(M)\det(CZ+D)^{r_{0}r}\varrho(CZ+D)f(Z). \end{equation*} $\varrho\colon \mathrm{GL}(n, \mathbb{C}) \to \mathrm{GL}(\mathcal{Z})$ is a rational representation on a finite dimensional complex vector space $\mathcal{Z}$. These spaces can be collected in a graded $A(\Gamma)$-module \begin{equation*} \mathcal{M} = \mathcal{M}_{\Gamma}(r_{0}, v, \varrho) := \bigoplus_{r \in \mathbb{Z}}[\Gamma, rr_{0}, v^{r}, \varrho]. \end{equation*} We treat in this paper some special cases in genus 2. The first one is essentially due to Wieber. Here the starting weight is $1/2$, the starting multiplier system is the multiplier system $v_{\Theta}$ and for $\varrho$ we take the second symmetric power of standard representation. Thus we consider a variant of this case and a new example. In this final case the starting weight is $1/2$, the starting multiplier system is the theta multiplier system $v_{\vartheta}$ and for $\varrho$ we take the standard representation. In all these cases we will determine the structure of $\mathcal{M}$.