Non-vanishing of central values of L-functions with angular restrictions

TL;DR

This paper investigates the non-vanishing of central values of L-functions with angular restrictions, employing trace function theory, mollification, and recent bilinear sum bounds to establish positive proportion non-vanishing results.

Contribution

It introduces a novel approach combining Katz's classification of sheaves and recent bilinear sum bounds to analyze L-functions with angular restrictions.

Findings

Proves positive proportion of non-vanishing central L-values under angular restrictions.

Employs trace function theory and mollification techniques effectively.

Provides new bounds on bilinear sums of trace functions.

Abstract

We study the angular restrictions for the second moment of toroidal families of $L$-functions using the general theory of trace functions. With the mollification technique we deduce non-vanishing of a positive proportion. Our two main ingredients are classification results of Katz to determine the sheaves at play and a recent result of Fouvry, Kowalski, Michel and Sawin to bound bilinear sums of their trace functions.