On the Spezialschar of Maass

TL;DR

This paper introduces a new subspace called the Spezialschar within Siegel modular forms, characterizes it in the degree 2 case, and relates it to special L-value non-vanishing, advancing understanding of Maass forms and their properties.

Contribution

It defines the Spezialschar for Siegel modular forms, proves its equivalence to Maass Spezialschar in degree three, and provides an embedding characterization for degree two forms.

Findings

Spezialschar is identified as the Maass Spezialschar in degree three.

An embedding of degree two forms into symmetric squares of forms of higher weight is established.

The work links the Spezialschar to non-vanishing of special L-values, with a key example provided.

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.