On analytic properties of the standard zeta function attached to a vector valued modular form

TL;DR

This paper establishes the meromorphic continuation and functional equation of the standard zeta function linked to vector valued modular forms, using integral representations involving Eisenstein series and Weil representations.

Contribution

It proves a Garrett-Böcherer decomposition for genus 2 vector valued Siegel Eisenstein series and demonstrates the analytic continuation and functional equation of the associated zeta function.

Findings

Meromorphic continuation of the zeta function to the entire complex plane

Functional equation satisfied by the zeta function

Decomposition of genus 2 Eisenstein series with Weil representation

Abstract

We proof a Garrett-B\"ocherer decomposition of a vector valued Siegel Eisenstein series $E_{l,0}^2$ of genus 2 transforming with the Weil representation of $\text{Sp}_2(\mathbb{Z})$ on the group ring $\mathbb{C}[(L'/L)^2]$. We show that the standard zeta function associated to a vector valued common eigenform $f$ for the Weil representation can be meromorphically continued to the whole $s$-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms of $f$ and $E_{l,0}^2$.