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

TL;DR

This paper proves that at least 77% of Artin L-functions associated with D4-quartic function fields, ordered by conductor, do not vanish at the central point, extending previous results over the rationals.

Contribution

It extends the non-vanishing results of Artin L-functions to D4-quartic function fields using low-lying zeros and one-level density methods.

Findings

At least 77% of these L-functions are non-vanishing at the central point.

Extended Rudnick's method from Dirichlet L-functions to D4-case.

Analyzed L-functions associated with D4-fields with large quadratic subfield discriminant.

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 Altu\u{g}, Shankar, Varma and Wilson.