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Non-vanishing of Artin $L$-functions associated with $D_4$-quartic function fields ordered by conductor

Victor Ahlquist

TL;DR

The paper analyzes the non-vanishing of Artin $L$-functions $P_{L/K}(s)=\zeta_L(s)/\zeta_K(s)$ attached to $D_4$-quartic function fields over $\mathbb{F}_q(T)$ ordered by conductor. It develops a detailed counting framework for $D_4$-quartic fields, including a refined breakdown into subfamilies by the quadratic subfield discriminant, and establishes precise asymptotics using the flipped-field method. The core technical engine is the one-level density for these families, handled separately for fields with small and large subfields, with careful control of rogue splitting types and character sums in the flipped field. By optimizing test functions in the explicit formula, the authors derive a rigorous lower bound of $77\%$ non-vanishing at the central point, and discuss extensions to analogous quadratic-quartic families over number fields, highlighting the broader impact on non-vanishing phenomena in Artin $L$-functions.

Abstract

We study the low-lying zeros of certain Artin $L$-functions associated with $D_4$-quartic function fields. Specifically, we prove that when ordered by conductor, at least $77\%$ of these $L$-functions are non-vanishing at the central point. This generalises and extends results over $\mathbb{Q}$ due to Durlanik, proving that an infinite number of these $L$-functions are non-vanishing. We obtain these results by examining the low-lying zeros of the $L$-functions using the one-level density. Specifically, we apply and extend a method used by Rudnick, who studied Dirichlet $L$-functions associated with quadratic function field extensions, to the $D_4$-case. The main difficulty is studying $L$-functions which are associated to $D_4$-fields whose quadratic subfield is of large discriminant. These $L$-functions are studied by utilising the so-called flipped field of a $D_4$ extension, combining a method introduced by Friedrichsen for counting $D_4$-fields, with explicit ramification theory in such fields provided by Altuğ, Shankar, Varma and Wilson.

Non-vanishing of Artin $L$-functions associated with $D_4$-quartic function fields ordered by conductor

TL;DR

The paper analyzes the non-vanishing of Artin -functions attached to -quartic function fields over ordered by conductor. It develops a detailed counting framework for -quartic fields, including a refined breakdown into subfamilies by the quadratic subfield discriminant, and establishes precise asymptotics using the flipped-field method. The core technical engine is the one-level density for these families, handled separately for fields with small and large subfields, with careful control of rogue splitting types and character sums in the flipped field. By optimizing test functions in the explicit formula, the authors derive a rigorous lower bound of non-vanishing at the central point, and discuss extensions to analogous quadratic-quartic families over number fields, highlighting the broader impact on non-vanishing phenomena in Artin -functions.

Abstract

We study the low-lying zeros of certain Artin -functions associated with -quartic function fields. Specifically, we prove that when ordered by conductor, at least of these -functions are non-vanishing at the central point. This generalises and extends results over due to Durlanik, proving that an infinite number of these -functions are non-vanishing. We obtain these results by examining the low-lying zeros of the -functions using the one-level density. Specifically, we apply and extend a method used by Rudnick, who studied Dirichlet -functions associated with quadratic function field extensions, to the -case. The main difficulty is studying -functions which are associated to -fields whose quadratic subfield is of large discriminant. These -functions are studied by utilising the so-called flipped field of a extension, combining a method introduced by Friedrichsen for counting -fields, with explicit ramification theory in such fields provided by Altuğ, Shankar, Varma and Wilson.

Paper Structure

This paper contains 29 sections, 19 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgements
  4. Conventions
  5. Preliminaries
  6. Function fields and $L$-functions
  7. The Galois theory of $D_4$-fields and Artin $L$-functions
  8. Counting $D_4$-quartic fields
  9. Counting the number of quadratic extensions
  10. Counting fields with
  11. Counting fields with
  12. The one-level density: fields with small subfields
  13. Class groups and Hecke characters
  14. Upper bounds on sums of divisors
  15. Splitting the summation and summing over $\mathcal{P}$
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $q$ be a prime power coprime to $2$ and larger than a sufficiently large absolute constant. Define with one representative from each isomorphism class. Then, at least $77 \%$ of the Artin $L$-functions $P_{L/K}(s):=\zeta_L(s)/\zeta_K(s)$ associated with $(L,K)\in \mathcal{F}(X)$ are non-vanishing at the central point $s=1/2$. More precisely,

Theorems & Definitions (28)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark
  • Proposition 1.4
  • Theorem 1.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 18 more