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