Non vanishing for cubic Hecke $L$-functions

I will discuss recent work with Alexander Dunn, Alexandre de Faveri and Joshua Stucky, in which we prove that a positive proportion of Hecke $L$-functions associated to the cubic residue symbol modulo squarefree Eisenstein integers do not vanish at the central point. Our principal new contribution is the asymptotic evaluation of the mollified second moment with power saving error term. No such asymptotic formula was previously known for a cubic family (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 Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish.