Spectral moment formulae for GL(3) × GL(2) L-functions I : The cuspidal case

Kwan, Chung-Hang

doi:10.2140/ant.2024.18.1817

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Spectral moment formulae for $\operatorname{GL}(3) \times \operatorname{GL}(2)$ $L$-functions I: The cuspidal case

Chung-Hang Kwan

Vol. 18 (2024), No. 10, 1817–1862

DOI: 10.2140/ant.2024.18.1817

Abstract

Spectral moment formulae of various shapes have proven very successful in studying the statistics of central $L$ -values. We establish, in a completely explicit fashion, such formulae for the family of $GL (3) \times GL (2)$ Rankin–Selberg $L$ -functions using the period integral method. Our argument does not rely on either the Kuznetsov or Voronoi formulae. We also prove the essential analytic properties and derive explicit formulae for the integral transform of our moment formulae. We hope that our method will provide deeper insights into moments of $L$ -functions for higher-rank groups.

Keywords

automorphic forms, automorphic $L$-functions, Maass forms, moments of $L$-functions, Rankin–Selberg $L$-functions, period integrals, Whittaker functions, hypergeometric functions, Poincare series

Mathematical Subject Classification

Primary: 11F55

Secondary: 11F72

Milestones

Received: 29 December 2021

Revised: 29 August 2023

Accepted: 12 October 2023

Published: 7 October 2024

Authors

Chung-Hang Kwan
Department of Mathematics University College London London United Kingdom

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Vol. 18, No. 10, 2024