Kloosterman Sums and a Mean Value for Dirichlet Polynomials

Watt, Nigel

doi:10.1006/jnth.1995.1086

Cited by 36 publications

(42 citation statements)

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“…Our results in [46] include analogues, for SL(2, C), of several of the (SL(2, R) related) results obtained by Deshouillers and Iwaniec in [9], and also the 'SL(2, C) analogue' of Theorem 2 of [45]. In work only partially written up we have made use of the results of [46] in estimating a particular type of weighted fourth power moment for the family of Hecke zeta functions ζ(s, λ k ) k∈Z given, for Re(s) > 1, by the cases of (1.8.7) in which χ : O → {1} and p is even.…”

supporting

confidence: 73%

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

Watt¹

2014

Journal of Number Theory

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We prove, in respect of an arbitrary Hecke congruence subgroup Γ = Γ 0 (q 0 ) ≤ 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 c = ∞). 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 P SL(2, Z[i])\P SL(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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confidence: 73%

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

Watt¹

2014

Journal of Number Theory

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“…We consider Theorem 9, which is an analogue of [21,Theorem 2], to be the foremost achievement of this paper. Indeed, both Theorems 8 and 9 are play a crucial part in a significant application that we will come to shortly (after some discussion of Theorem 8, and of the proof of Theorem 9).…”

Section: Outline Of Results and Methodsmentioning

confidence: 99%

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

Watt¹

2013

Funct. Approx. Comment. Math.

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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 Γ ≤ SL(2, Z[i]), and correspond to exceptional eigenvalues of the Laplace operator on the space L 2 (Γ\SL(2, 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 P SL(2, Z[i])\P SL(2, 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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“…Our method is an adaptation to algebraic number fields of the sieve method introduced by the first author [8,9] and shares many features with [2] and [3]. Our result is not quite as strong as the result in [3] because we do not have an analogue for Hecke L-functions of Watt's mean-value theorem [18].…”

Section: The Implied Constant Depends Only On Kmentioning

confidence: 83%