PRODUCT OF RANKIN-SELBERG CONVOLUTIONS AND A NEW PROOF OF JACQUET’S LOCAL CONVERSE CONJECTURE

Source abstract

Abstract This article constructs a family of integrals representing the product of Rankin–Selberg L -functions for GL l × GL m ${\mathrm {GL}}_l \times {\mathrm {GL}}_m$ upper G upper L Subscript l Baseline times upper G upper L Subscript m and GL l × GL n ${\mathrm {GL}}_l \times {\mathrm {GL}}_n$ upper G upper L Subscript l Baseline times upper G upper L Subscript n in the regime m + n &lt; l $m+n<l$ m plus n less than l . When n = 0 $n=0$ n equals 0 , these integrals reduce (up to a shift) to the classical Rankin-Selberg convolution integrals of Jacquet–Piatetski-Shapiro–Shalika (JPSS), thereby generalizing the JPSS integrals. We establish their basic properties and use them to define local gamma factors. As an application, we provide a new proof of Jacquet’s local converse conjecture.