The Ranking-Selberg integral on ${\bf GSp(2)}$ for square free levels

TL;DR

This paper explicitly computes a Rankin-Selberg integral for vector-valued Siegel cusp forms of square-free levels, providing detailed local zeta-integral evaluations.

Contribution

It offers explicit calculations of the Rankin-Selberg integral and local zeta-integrals for Siegel cusp forms at square-free levels, advancing understanding of these automorphic forms.

Findings

Explicit evaluation of the Rankin-Selberg integral for GSp(2)

Precise local zeta-integral computations for specific test functions

Enhanced understanding of Siegel cusp forms at square-free levels

Abstract

We explicitly compute the Rankin-Selberg type integral introduced by Piatetski-Shapiro over adeles for vector-valued Siegel cusp forms of square-free levels $\Gamma_0(N)$. On the way, for particular test functions in the Bessel models of irreducible admissible representations, exact evaluations of the local zeta-integrals are given.