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June 2013 Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$
Nigel Watt
Funct. Approx. Comment. Math. 48(2): 213-376 (June 2013). DOI: 10.7169/facm/2013.48.2.4

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 $\Gamma\leq SL(2,{\mathbb Z}[i])$, and correspond to exceptional eigenvalues of the Laplace operator on the space $L^2(\Gamma\backslash SL(2,{\mathbb C})/SU(2))$. 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 $PSL(2,{\mathbb Z}[i])\backslash PSL(2,{\mathbb C})$. 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).

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Nigel Watt. "Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$." Funct. Approx. Comment. Math. 48 (2) 213 - 376, June 2013. https://doi.org/10.7169/facm/2013.48.2.4

Information

Published: June 2013
First available in Project Euclid: 18 June 2013

zbMATH: 1277.11041
MathSciNet: MR3100141
Digital Object Identifier: 10.7169/facm/2013.48.2.4

Subjects:
Primary: 11F30

Rights: Copyright © 2013 Adam Mickiewicz University

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Vol.48 • No. 2 • June 2013
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