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

Abstract

We show that a positive proportion of the values $L(1/2,χ_c)$ are non-zero, where $χ_c$ 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 $L(1/2,χ_c)\neq 0$, 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 $χ$ of (prime) order $\ell$ such that $L(1/2, χ) \neq 0$ (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.