Functiones et Approximatio Commentarii Mathematici

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

Nigel Watt

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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

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

First available in Project Euclid: 18 June 2013

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Zentralblatt MATH identifier

Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F37: Forms of half-integer weight; nonholomorphic modular forms 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula 11L05: Gauss and Kloosterman sums; generalizations 11L07: Estimates on exponential sums 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11N13: Primes in progressions [See also 11B25] 11N35: Sieves 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11R44: Distribution of prime ideals [See also 11N05] 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 44A15: Special transforms (Legendre, Hilbert, etc.)

spectral theory large sieve mean value Hecke congruence group Gaussian number field Gaussian integers sum formula automorphic form cusp form non-holomorphic modular form Fourier coefficient Kloosterman sum inverse Bessel transform eigenvalue conjecture grössencharakter Hecke character


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.

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