Integrality and cuspidality of pullbacks of nearly holomorphic Siegel Eisenstein series

Abstract

We study nearly holomorphic Siegel Eisenstein series of general levels and characters on ${ℍ}_{2n}$, 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 ${ℍ}_n\times {ℍ}_n$ are cuspidal under certain assumptions.