Congruences for dimensions of spaces of Siegel cusp forms and $4$-core partitions

TL;DR

This paper explores congruences linking the dimensions of Siegel cusp form spaces, class numbers of quadratic fields, and 4-core partition functions, revealing new relationships in number theory.

Contribution

It establishes novel congruences connecting Siegel cusp form dimensions, class numbers, and 4-core partitions using automorphic representation theory.

Findings

Derived congruences between Siegel cusp form dimensions and class numbers.

Established congruences involving 4-core partition function and Siegel cusp forms.

Connected automorphic representations with classical partition functions.

Abstract

Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core partition function $c_4(n)$ and dimensions of spaces of Siegel cusp forms of degree $2$.