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

Filippo Berta, Svenja zur Verth

TL;DR

The paper analyzes angular restrictions for central L-values in toroidal Dirichlet-character families by combining the approximate functional equation with trace-function technology. It reduces key bilinear sums to hypergeometric trace functions and leverages Katz’s monodromy classification together with FKMS bilinear bounds to achieve cancellation in a broad class of sums, including non-gallant exceptions treated via Poisson summation. After establishing asymptotic formulae for the twisted and untwisted second moments, it introduces mollifiers to prove that a positive proportion of primitive characters in a prescribed angular sector yield simultaneously non-vanishing central values at $s= frac12$ for $L( ext{χ}^a, frac12)L( ext{χ}^b, frac12)$. The results extend prior toroidal moment analyses by incorporating angular restrictions and provide explicit constants and effective bounds, with potential applications to subfamilies and related $L$-function families.

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.

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

TL;DR

The paper analyzes angular restrictions for central L-values in toroidal Dirichlet-character families by combining the approximate functional equation with trace-function technology. It reduces key bilinear sums to hypergeometric trace functions and leverages Katz’s monodromy classification together with FKMS bilinear bounds to achieve cancellation in a broad class of sums, including non-gallant exceptions treated via Poisson summation. After establishing asymptotic formulae for the twisted and untwisted second moments, it introduces mollifiers to prove that a positive proportion of primitive characters in a prescribed angular sector yield simultaneously non-vanishing central values at for . The results extend prior toroidal moment analyses by incorporating angular restrictions and provide explicit constants and effective bounds, with potential applications to subfamilies and related -function families.

Abstract

We study the angular restrictions for the second moment of toroidal families of -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.
Paper Structure (11 sections, 29 theorems, 160 equations)

This paper contains 11 sections, 29 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. The approximate functional equation
  3. Reduction to hypergeometric sums
  4. Dyadic partition and Pólya-Vinogradov method
  5. The special case $k=1$
  6. Bounds for trace functions of gallant sheaves
  7. Being gallant
  8. Katz classification
  9. Special cases
  10. Proof of the asymptotic formulae
  11. Mollifiers and non-vanishing

Key Result

Theorem 1.1

Let $I \subset (-\pi,\pi]$ be a non-empty interval and let $a,b$ be two non-zero integers. For any prime $q$ we define Then there exists a constant $C(a,b,I)>0$ so that for $q\to \infty$.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • ...and 58 more