A remark on $p$-adic Siegel Eisenstein series

TL;DR

This paper generalizes Serre's $p$-adic Eisenstein series for Siegel modular forms and proves their coincidence with certain genus theta series linked to quaternary quadratic forms.

Contribution

It introduces a $p$-adic Siegel Eisenstein series framework and establishes its equivalence with genus theta series for the first time.

Findings

Proved the coincidence between $p$-adic Eisenstein series and genus theta series.

Extended Serre's $p$-adic Eisenstein series to the Siegel modular form context.

Established new links between $p$-adic analysis and quadratic forms.

Abstract

A generalization of Serre's $p$-adic Eisenstein series in the case of Siegel modular forms is studied and a coincidence between a $p$-adic Siegel Eisenstein series and a genus theta series associated with a quaternary quadratic form is proved.