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

Abstract

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.

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

“…Through this one defines the function J ω ϕ ℓ,q (ν V , p V ) : G → C when (ν V , p V ) are the spectral parameters of an arbitrary irreducible subspace V ⊂ 0 L 2 (Γ\G). As noted in [22,Subsection 1.7] (see, in particular, [22,Relations (1.7.10) and (1.5.17)]), each term F c ω f in the Fourier expansion at any cusp c of any function f ∈ V K,ℓ,q is a constant multiple of the corresponding Jacquet integral, J ω ϕ ℓ,q (ν V , p V ) : G → C. Indeed, it is even possible to choose, for the subspace factors V K,ℓ,q in (1.1.5), a system of generators,…”

“…In the paper [22] (to appear) the case F = Q(i) of the spectral to Kloosterman formula for Γ 0 (q) obtained in [19] is slightly generalised, so as to apply for arbitrary pairs of cusps a, b (rather than just for a = b = ∞); by means of that generalised formula, and bounds for the relevant generalised Kloosterman sums, we obtain, in [22,Theorem 1], the spectral large sieve inequality which is reproduced as 'Theorem 2' in this paper. In the present paper it is instead the Kloosterman to spectral summation formula, Theorem 1 below, that has the more prominent part to play (though we use the spectral to Kloosterman summation formula in proving Theorem 11).…”

“…The condition Re(ν) > 1 ensures absolute convergence of the sum in (1.1.12): this, and more delicate issues of convergence, are discussed in [22,Subsection 1.8] (but see also [7,Chapter 3], [4,Section 5] or [19,Section 3.3]). Here it suffices to record that the Eisenstein series given by (1.1.12) are infinitely differentiable Γ-automorphic functions on G, and inherit from ϕ ℓ,q (ν, p) the property of being eigenfunctions of both Casimir operators Ω ± with corresponding eigenvalues…”

“…This theorem is a minor extension of Lokvenec-Guleska's result [19,Theorem 12.3.2], which applies only to the case a = b = ∞ (though being, in other important respects considerably more general than our theorem). The proof is a straightforward application of [22 …”

“…To prove our theorem we need only verify that, for some σ > 1/2, the function h = Kf satisfies the hypotheses (i)-(iii) of [22,Theorem B]: for then the equation (1.2.1) follows by the direct use of (1.2.6) and (1.2.5) to effect appropriate substitutions in the case h = Kf of [22, Theorem B, Equation (1.9.1)]. Those hypotheses are satisfied by h = Kf if, when S σ = {ν ∈ C : |Re(ν)| ≤ σ}, one has all of the following:…”