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arXiv · 1302.3112

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

Nigel Watt
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Abstract

We prove, in respect of an arbitrary Hecke congruence subgroup Γ=Γ_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 Γ-automorphic cusp forms on SL(2,C). The Fourier coefficients in question may arise from the Fourier expansion at any given cusp c of Γ: 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 Γ\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.

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Nigel Watt. 2013-02-13. Spectral large sieve inequalities for Hecke congruence subgroups of SL(2,Z[i]). https://doi.org/10.1016/j.jnt.2014.01.018

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