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Wide moments of $L$-functions I: Twists by class group characters of imaginary quadratic fields

Asbjørn Christian Nordentoft

Vol. 18 (2024), No. 4, 735–770
DOI: 10.2140/ant.2024.18.735

We calculate certain “wide moments” of central values of Rankin–Selberg L-functions L(π Ω, 1 2) where π is a cuspidal automorphic representation of GL 2 over and Ω is a Hecke character (of conductor 1) of an imaginary quadratic field. This moment calculation is applied to obtain “weak simultaneous” nonvanishing results, which are nonvanishing results for different Rankin–Selberg L-functions where the product of the twists is trivial.

The proof relies on relating the wide moments of L-functions to the usual moments of automorphic forms evaluated at Heegner points using Waldspurger’s formula. To achieve this, a classical version of Waldspurger’s formula for general weight automorphic forms is derived, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error terms), together with nonvanishing results for certain period integrals. In particular, we develop a soft technique for obtaining the nonvanishing of triple convolution L-functions.

moments of $L$-functions, periods of automorphic forms
Mathematical Subject Classification
Primary: 11F67
Secondary: 11M41
Received: 17 January 2022
Revised: 30 January 2023
Accepted: 13 May 2023
Published: 26 February 2024
Asbjørn Christian Nordentoft
Institut Galilée

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