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

TL;DR

This paper establishes new spectral large sieve inequalities for Fourier coefficients of automorphic cusp forms on SL(2,C) related to Hecke congruence subgroups of SL(2,Z[i]), extending the sum formula to arbitrary cusps.

Contribution

It extends the spectral-Kloosterman sum formula to arbitrary cusps and derives new bounds for Fourier coefficients of automorphic forms on SL(2,C).

Findings

New upper bounds for Fourier coefficients of automorphic cusp forms.

Extension of spectral-Kloosterman sum formula to all cusps.

Proof of the extended sum formula included.

Abstract

We prove, in respect of an arbitrary Hecke congruence subgroup \Gamma =\Gamma_0(q_0) of the group SL(2,Z[i]), some new upper bounds (or `spectral large sieve inequalities') for sums involving Fourier coefficients of \Gamma -automorphic cusp forms on SL(2,C). The Fourier coefficients in question may arise from the Fourier expansion at any given cusp c of \Gamma : our results are not limited to the case in which c is the cusp at infinity. For this reason, our proof is reliant upon an extension, to arbitrary cusps, of the spectral-Kloosterman sum formula for \Gamma\SL(2,C) obtained by Hristina Lokvenec-Guleska in her doctoral thesis (generalising the sum formulae of Roelof Bruggeman and Yoichi Motohashi for PSL(2,Z[i])\PSL(2,C) in several respects, though not as regards the choice of cusps). A proof of the required extension of the sum formula is given in an appendix.