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Kwan, Chung-Hang

Spectral moment formulae for \(\mathrm{GL}(3)\times\mathrm{GL}(2)\) \(L\)-functions. I: The cuspidal case. (English) Zbl 07929055

Algebra Number Theory 18, No. 10, 1817-1862 (2024).
Summary: 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 \(\mathrm{GL}(3)\times\mathrm{GL}(2)\) 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.

MSC:

11F55 Other groups and their modular and automorphic forms (several variables)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)

Keywords:

automorphic forms; automorphic \(L\)-functions; Maass forms; moments of \(L\)-functions; Rankin-Selberg \(L\)-functions; period integrals; Whittaker functions; hypergeometric functions; Poincaré series
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References:

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