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

Abstract

Abstract This article constructs a family of integrals representing the product of Rankin–Selberg L -functions for <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>GL</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>l</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>GL</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>m</mml:mi> </mml:msub> </mml:math> ${\mathrm {GL}}_l \times {\mathrm {GL}}_m$ upper G upper L Subscript l Baseline times upper G upper L Subscript m and <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>GL</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>l</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>GL</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:math> ${\mathrm {GL}}_l \times {\mathrm {GL}}_n$ upper G upper L Subscript l Baseline times upper G upper L Subscript n in the regime <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>l</mml:mi> </mml:math> $m+n&lt;l$ m plus n less than l . When <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> $n=0$ 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.