Dimension formulas for Siegel modular forms of level $4$

TL;DR

This paper derives new dimension formulas for spaces of degree 2 Siegel modular forms of level 4, connecting automorphic representations and local representation theory to explicit dimension counts.

Contribution

It provides novel dimension formulas for Siegel modular forms of level 4, especially for Klingen subgroups, using automorphic representation theory and local fixed vector dimensions.

Findings

New dimension formulas for Siegel modular forms at level 4.

Explicit counts of automorphic representations with specific local properties.

Connections between local representation theory and global modular form dimensions.

Abstract

We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree $2$ with respect to certain congruence subgroups of level $4$. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A})$ whose local component at $p=2$ admits non-zero fixed vectors under the principal congruence subgroup of level $2$. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at $p=2$, we obtain formulas for the number of relevant automorphic representations. These in turn lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level $4$.