Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions I: The Cuspidal Case

TL;DR

This paper derives explicit spectral moment formulae for $GL(3) imes GL(2)$ Rankin-Selberg $L$-functions using period integrals, avoiding Kuznetsov and Voronoi formulas, aiming to inform higher-rank $L$-function studies.

Contribution

The authors develop a new explicit method for spectral moment formulae of $GL(3) imes GL(2)$ $L$-functions that bypasses traditional formulas, providing a foundation for higher-rank analysis.

Findings

Established explicit spectral moment formulae for $GL(3) imes GL(2)$ $L$-functions.

Proved analytic properties and explicit integral transforms of the moment formulae.

Method avoids Kuznetsov and Voronoi formulas, simplifying the analysis.

Abstract

Spectral moment formulae of various shapes have proven to be very successful in studying the statistics of central $L$-values. In this article, 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. The Kuznetsov and the Voronoi formulae are not needed in our argument. We also prove the essential analytic properties and explicit formulae for the integral transform of our moment formulae. It is hoped that our method will provide insights into moments of $L$-functions for higher-rank groups.