On $p$-adic Siegel--Eisenstein series II: How to avoid the regularity condition for $p$

TL;DR

This paper demonstrates that the regularity condition on prime p can be removed for certain low-degree p-adic Siegel--Eisenstein series, expanding their applicability beyond regular primes.

Contribution

It extends previous results by removing the regularity condition on p for low-degree cases where n ≤ 2k+1.

Findings

Regularity condition on p is unnecessary for n ≤ 2k+1.

p-adic Siegel--Eisenstein series coincide with classical modular forms without regularity assumption.

Broader applicability of these series to non-regular primes in low-degree cases.

Abstract

In a previous paper, the authors showed that two kinds of $p$-adic Siegel--Eisenstein series of degree $n$ coincide with classical modular forms of weight $k$ for $\Gamma _0(p)$, under the assumption that $p$ is a regular prime. The purpose of this paper is to show that this condition on $p$ can be removed if the degree $n$ is low compared with $k$, namely, $n\le 2k+1$.