Non-vanishing for cubic Hecke $L$-functions

Abstract: 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 previously known (even over function fields). 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.