One-Level Density and Non-Vanishing for Cubic L-Functions Over the Eisenstein Field

Abstract

Abstract We study the one-level density for families of $L$-functions associated with cubic Dirichlet characters defined over the Eisenstein field. We show that the family of $L$-functions associated with the cubic residue symbols $\chi _n$ with $n$ square-free and congruent to 1 modulo 9 satisfies the Katz–Sarnak conjecture for all test functions whose Fourier transforms are supported in $(-13/11, 13/11)$, under the Generalized Riemann Hypothesis. This is the first result extending the support outside the trivial range$(-1, 1)$ for a family of cubic $L$-functions. This implies that a positive density of the $L$-functions associated with these characters do not vanish at the central point $s=1/2$. A key ingredient in our proof is a bound on an average of generalized cubic Gauss sums at prime arguments, whose proof is based on the work of Heath-Brown and Patterson [22, 23].