Integrality and cuspidality of pullbacks of nearly holomorphic Siegel Eisenstein series

TL;DR

This paper investigates the algebraic properties and cuspidality of pullbacks of nearly holomorphic Siegel Eisenstein series, establishing integrality of Fourier coefficients and conditions for cuspidality in the context of Siegel modular forms.

Contribution

It proves the integrality of Fourier coefficients of these Eisenstein series and identifies conditions under which their pullbacks are cuspidal.

Findings

Fourier coefficients lie in the ring of integers of $Q_p$ for large primes p.

Pullbacks to $H_n imes H_n$ are cuspidal under certain assumptions.

Provides new insights into the arithmetic and geometric properties of Siegel Eisenstein series.

Abstract

We study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions.