Weighted spectral large sieve inequalities for Hecke congruence subgroups of SL(2,Z[i])

Abstract

We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a Hecke congruence subgroup Γ=Γ_0(q) of the group SL(2,Z[i]), and correspond to exceptional eigenvalues of the Laplace operator on the space L^2(Γ\SL(2,C)/SU(2)). These results are, for certain applications, an effective substitute for the generalised Selberg eigenvalue conjecture. We give a proof of one such application, which is an upper bound for a sum of generalised Kloosterman sums (of significance in the study of certain mean values of Hecke zeta-functions with groessencharakters). Our proofs make extensive use of Lokvenec-Guleska's generalisation of the Bruggeman-Motohashi summation formulae for PSL(2,Z[i])\PSL(2,C). We also employ a bound of Kim and Shahidi for the first eigenvalues of the relevant Laplace operators, and an `unweighted' spectral large sieve inequality (our proof of which is to appear separately).