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

TL;DR

This paper establishes new bounds for sums involving Fourier coefficients of automorphic cusp forms related to Hecke congruence subgroups of SL(2,Z[i]), providing effective tools for studying eigenvalues and Kloosterman sums in number theory.

Contribution

It introduces novel bounds for weighted mean values of Fourier coefficients of automorphic forms on SL(2,Z[i]) and applies these to bound sums of Kloosterman sums, advancing understanding of eigenvalues and spectral analysis.

Findings

New bounds for Fourier coefficient sums of automorphic cusp forms.

Effective upper bounds for sums of Kloosterman sums.

Application of spectral large sieve inequalities to number theory problems.

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 \Gamma =\Gamma_0(q) of the group SL(2,Z[i]), and correspond to exceptional eigenvalues of the Laplace operator on the space L^2(\Gamma\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).