Non-vanishing for cubic $L$--functions

TL;DR

This paper demonstrates that a positive proportion of cubic $L$-functions over function fields do not vanish at the critical point, using mollified moments and advanced analytic techniques, with results applicable in various settings.

Contribution

It provides the first positive proportion non-vanishing result for cubic $L$-functions over function fields using mollified moments and sharp bounds, extending previous work.

Findings

Proves positive proportion of non-vanishing cubic $L$-functions over $_q[T]$

Develops sharp upper bounds for mollified second moments

Results are explicit but with very small positive proportion

Abstract

We prove that there is a positive proportion of $L$-functions associated to cubic characters over $\mathbb{F}_q[T]$ that do not vanish at the critical point $s=1/2$. This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic $L$-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester-Radziwill, which in turn develops further ideas from the work of Soundararajan, Harper, and Radziwill-Soundararajan. We work in the non-Kummer setting when $q \equiv 2\pmod{3}$, but our results could be translated into the Kummer setting when $q\equiv 1\pmod{3}$ as well as into the number field case (assuming the Generalized Riemann Hypothesis). Our positive proportion of non-vanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.