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

TL;DR

This paper develops a method to compute wide moments of Rankin--Selberg L-functions associated with automorphic forms and imaginary quadratic fields, leading to new non-vanishing results and a classical Waldspurger formula extension.

Contribution

It introduces a novel approach to evaluate wide moments of L-functions and derives a classical Waldspurger formula for general weight automorphic forms.

Findings

Calculated wide moments of L-functions using Heegner points.

Established non-vanishing results for products of L-functions.

Extended Waldspurger's formula to general weights.

Abstract

We calculate certain "wide moments" of central values of Rankin--Selberg $L$-functions $L(\pi\otimes \Omega, 1/2)$ where $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_2$ over $\mathbb{Q}$ and $\Omega$ is a Hecke character (of conductor $1$) of an imaginary quadratic field. This moment calculation is applied to obtain "weak simultaneous" non-vanishing results, which are non-vanishing results for different Rankin--Selberg $L$-functions where the product of the twists is trivial. The proof relies on relating the wide moments 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 non-vanishing results for certain period integrals. In particular, we develop a soft technique for obtaining non-vanishing of triple convolution $L$-functions.