Moments of Central $L$-values for Maass Forms over Imaginary Quadratic Fields

Abstract

In this paper, over imaginary quadratic fields, we consider the family of $L$-functions $L(s,f)$ for an orthonormal basis of spherical Hecke--Maass forms $f$ with Archimedean parameter $t_f$. We establish asymptotic formulae for the twisted first and second moments of the central values $L(\frac{1}{2},f)$, which can be applied to prove that at least $33\%$ of $L(\frac{1}{2},f)$ with $t_f⩽T$ are non-vanishing as $T\to \infty$. Our main tools are the spherical Kuznetsov trace formula and the Voronoï summation formula over imaginary quadratic fields.

28 pages. To appear in Trans Amer. Math. Soc