On Siegel--Eisenstein series of level $p$ and their $p$-adic properties

TL;DR

This paper constructs specific level $p$ Siegel--Eisenstein series with quadratic characters, explicitly computes their Fourier coefficients, and demonstrates their $p$-adic limit properties, extending to nonquadratic characters as well.

Contribution

It introduces a new construction of level $p$ Siegel--Eisenstein series with quadratic characters and proves their $p$-adic convergence, also extending results to nonquadratic characters.

Findings

Constructed a level $p$ Siegel--Eisenstein series with quadratic character

Explicitly calculated Fourier coefficients of the series

Proved the series is a $p$-adic limit of level 1 Eisenstein series

Abstract

We construct a Siegel--Eisenstein series of level $p$ with a quadratic character mod $p$ which is a $U(p)$-eigenfunction with eigenvalue $1$, and calculate its Fourier coefficients explicitly. We show that this Siegel--Eisenstein series is a $p$-adic Siegel--Eisenstein series, i.e., it is a $p$-adic limit of a sequence of Siegel--Eisenstein series of level $1$. We prove also that the Siegel--Eisenstein series with a nonquadratic character mod $p$ constructed by Takemori is also a $p$-adic Siegel--Eisenstein series.