Generic level $p$ Eisenstein congrunces for GSp$_4$

TL;DR

This paper explores level p Eisenstein congruences for GSp4, linking automorphic and Galois representations to establish conditions for congruences, supporting previous computational findings.

Contribution

It generalizes Harder's level 1 congruences to level p for GSp4 and provides theoretical conditions for their existence.

Findings

Conditions for the existence of paramodular forms satisfying congruences

Theoretical justification for computational evidence

Extension of Harder's conjectures to level p

Abstract

We investigate level $p$ Eisenstein congruences for GSp$_4$, generalisations of level $1$ congruences predicted by Harder. By studying the associated Galois and automorphic representations we see conditions that guarantee the existence of a paramodular form satisfying the congruence. This provides theoretical justification for computational evidence found in the author's previous paper.