On the Spezialschar of Maass

Abstract

Let $M_k^{(n)}$ be the space of Siegel modular forms of degree $n$ and even weight $k$. In this paper firstly a certain subspace $\mathsf{Spez}(M_k^{(2n)})$ the Spezialschar of $M_k^{(2n)}$ is introduced. In the setting of the Siegel three-fold it is proven that this Spezialschar is the Maass Spezialschar. Secondly an embedding of $M_k^{(2)}$ into a direct sum $\oplus_{\nu = 0}^{\lfloor \frac{k}{10} \rfloor} \text{Sym}^2 M_{k + 2 \nu}$ is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the non-vanishing of certain special values of L-functions related to the Gross-Prasad conjecture. This is illustrated by a significant example in the paper.