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

Nordentoft, Asbjørn Christian

doi:10.2140/ant.2024.18.735

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

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

Abstract

We calculate certain “wide moments” of central values of Rankin–Selberg $L$ -functions $L (π \otimes Ω, \frac{1}{2})$ where $π$ is a cuspidal automorphic representation of ${GL}_{2}$ over $ℚ$ 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.

Keywords

moments of $L$-functions, periods of automorphic forms

Mathematical Subject Classification

Primary: 11F67

Secondary: 11M41

Milestones

Received: 17 January 2022

Revised: 30 January 2023

Accepted: 13 May 2023

Published: 26 February 2024

Authors

Asbjørn Christian Nordentoft
LAGA Institut Galilée Villetaneuse France

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 18, No. 4, 2024