Product of Rankin-Selberg convolutions and a new proof of Jacquet's local converse conjecture

TL;DR

This paper introduces a new family of integrals representing products of Rankin-Selberg L-functions, generalizing previous constructions, and uses them to provide a novel proof of Jacquet's local converse conjecture.

Contribution

The authors construct a new family of integrals for Rankin-Selberg convolutions and apply them to prove Jacquet's local converse conjecture in a new way.

Findings

New integrals generalize classical Rankin-Selberg integrals.

Defined local gamma factors using the new integrals.

Provided a new proof of Jacquet's local converse conjecture.

Abstract

In this article, we construct a family of integrals which represent the product of Rankin-Selberg $L$-functions of $\mathrm{GL}_{l}\times \mathrm{GL}_m$ and of $\mathrm{GL}_{l}\times \mathrm{GL}_n $ when $m+n<l$. When $n=0$, these integrals are those defined by Jacquet--Piatetski-Shapiro--Shalika up to a shift. In this sense, these new integrals generalize Jacquet--Piatetski-Shapiro--Shalika's Rankin-Selberg convolution integrals. We study basic properties of these integrals. In particular, we define local gamma factors using this new family of integrals. As an application, we obtain a new proof of Jacquet's local converse conjecture using these new integrals.