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

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.