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

TL;DR

The paper addresses removing the $p$-regularity assumption in the identification of $p$-adic Siegel--Eisenstein limits with classical Siegel modular forms for $\Gamma_0(p)$ when the degree satisfies $n\le 2k+1$. It leverages mod $p^m$ singular modular forms and Bo-Ki techniques to relate the $p$-adic limits $\widetilde{E}^{(n)}_{(k,a)}$ to genus-theta expansions. The main result expresses the $p$-adic limit $\widetilde{E}_{{\boldsymbol k}_j}^{(n)}$ as a finite linear combination of genus theta-series $(\Theta^{(n)}_{{\rm gen}(S)})^0$ with level$(S)\mid p$ and characters $\chi_S=\chi_p^j$, with coefficients obtained as $m\to\infty$ limits of Fourier data. This extends prior work by removing the $p$-regularity constraint for small $n$ and provides a concrete $p$-adic decomposition that respects $U(p)$-invariance and the genus-theoretic structure.

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 $Γ_0(p)$, 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, $n\le 2k+1$.

Theorems & Definitions (10)

• Lemma 2.1: Bo-Ki2 Lemma 4.1
• Definition 2.2
• Theorem 2.3: Bo-Ki3
• Proposition 2.4: Bo-Ki1
• Remark 2.5
• Lemma 2.6
• proof
• Theorem 3.1
• Remark 3.2
• proof : Proof of Theorem \ref{['thm:main']}