## Functiones et Approximatio Commentarii Mathematici

### Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$

Nigel Watt

#### 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).

#### Article information

Source
Funct. Approx. Comment. Math., Volume 48, Number 2 (2013), 213-376.

Dates
First available in Project Euclid: 18 June 2013

https://projecteuclid.org/euclid.facm/1371586157

Digital Object Identifier
doi:10.7169/facm/2013.48.2.4

Mathematical Reviews number (MathSciNet)
MR3100141

Zentralblatt MATH identifier
1277.11041

#### Citation

Watt, Nigel. Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$. Funct. Approx. Comment. Math. 48 (2013), no. 2, 213--376. doi:10.7169/facm/2013.48.2.4. https://projecteuclid.org/euclid.facm/1371586157

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