Pullback formula for vector valued Siegel modular forms and its applications

TL;DR

This paper derives a pullback formula for vector valued Siegel Eisenstein series using a differential operator, and explores its applications to L-functions, Eisenstein series, and algebraicity of Siegel modular forms.

Contribution

It introduces a new pullback formula for vector valued Siegel Eisenstein series using Ibukiyama's differential operator, with applications to L-functions and algebraicity results.

Findings

Derived a pullback formula for vector valued Eisenstein series.

Applied the formula to study properties of L-functions and Eisenstein series.

Established algebraicity results for Siegel modular forms.

Abstract

Let $E_n$ be the Siegel Eisenstein series of degree $n$ and weight $k$ with a complex parameter $s$. In this paper, using a differential operator $D$ by Ibukiyama which sends a scalar valued Siegel modular form to the tensor product of two vector valued Siegel modular forms, under a certain condition, we give a formula of $DE_{p+q}$ on $H_p\times H_q$, where $H_n$ is the Siegel upper half space of degree $n$. Furthermore, we give some applications of this formula, i.e., analytic properies of standard $L$-functions and the Klingen Eisenstein series and algebraicity results for Siegel modular forms and standard $L$-functions.