Nonvanishing of $L$--functions associated to fixed order characters over function fields

TL;DR

This paper proves that a positive proportion of $L$-functions associated with fixed order characters over function fields do not vanish at the critical point, using advanced density and equidistribution techniques.

Contribution

It introduces new methods to show positive non-vanishing proportions for $L$-functions of fixed order characters, surpassing previous barriers and extending results to broader cases.

Findings

At least 1/6 of $L(1/2,\chi_c)$ are non-zero for cubic characters.

Established positive proportions of non-vanishing $L$-values for characters of order greater than 3.

Proved equidistribution of angles of shifted Gauss sums over primes.

Abstract

We show that a positive proportion of the values $L(1/2,\chi_c)$ are non-zero, where $\chi_c$ is the $\ell^{\text{th}}$ residue symbol for $\ell \geq 3$ over $\mathbb{F}_q[t]$, when averaging over square-free polynomials $c$ in $\mathbb{F}_q[t]$, as $q \equiv 1(\textrm{mod}\,{2\ell})$ is fixed and the degree of $c$ goes to infinity. In the case of $\ell=3$, we show that at least $1/6$ of $L(1/2,\chi_c)\neq 0$, while for $\ell>3$, the proportion depends on the order of the character. This improves a previous result of Ellenberg, Li, and Shusterman showing that there are infinitely many $\chi$ of (prime) order $\ell$ such that $L(1/2, \chi) \neq 0$ (with completely different techniques). Our result is achieved by computing the one-level density of zeros in the family of $L$--functions and surpassing the $(-1,1)$ barrier for the support of the Fourier transform of the test function, necessary to obtain a positive proportion of non-vanishing result. Using similar techniques, we also prove a result towards the equidistribution of the angles of the order $\ell$ shifted Gauss sums when summing over prime arguments, a result which may be of independent interest.