Non-vanishing for cubic Hecke $L$-functions

TL;DR

This paper proves that a positive proportion of cubic Hecke L-functions over the Eisenstein quadratic field do not vanish at the central point, using novel asymptotic analysis of mollified second moments and advanced analytic techniques.

Contribution

It provides the first asymptotic evaluation of the mollified second moment for a cubic family of L-functions, overcoming challenges posed by the non-orthogonal family structure.

Findings

A positive proportion of cubic Hecke L-functions do not vanish at the central point.

Established a Lindelöf-on-average upper bound for cubic Dirichlet series.

Developed new methods to handle non-orthogonal families in L-function non-vanishing proofs.

Abstract

Let $\omega$ be a primitive cubic root of unity. We study the non-vanishing problem for the family of Hecke $L$-functions associated to primitive cubic characters defined over the Eisenstein quadratic number field $\mathbb{Q}(\omega)$. We prove unconditionally that a positive proportion of Hecke $L$-functions associated to the cubic residue symbols $\chi_q$ with $q \in \mathbb{Z}[\omega]$ squarefree and $q \equiv 1 \pmod{9}$ do not vanish at the central point. Our proof goes through the method of first and second mollified moments. The principal new contribution of this paper is the asymptotic evaluation of the mollified second moment with power saving error term. No asymptotic formula for the mollified second moment of a cubic family was known (even over function fields) prior to the writing of this paper. Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindel\"{o}f-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the (unitary) family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet $L$-functions in the literature for which unconditional results are known: the symplectic family of quadratic characters and the unitary family of all Dirichlet characters $\chi \pmod{q}$. Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.