The Spectral Kuznetsov Formula on $SL(3)$

Abstract

Abstract. The SL(3) Kuznetsov formula exists in several versions, and has been employed with some success to study automorphic forms on SL(3). In each version, the weight functions on the geometric side are given by multiple integrals with complicated oscillating factors; this is the primary obstruction to its use. By describing them as solutions to systems of differential equations, we give power series and Mellin-Barnes integral representations of minimal dimension for these weight functions. This completes the role of harmonic analysis on symmetric spaces on the geometric side of the Kuznetsov formula, so that further study may be done through classical analytic techniques.The classical Kuznetsov formulas serve as a bridge between Fourier coefficients of Maass forms and Kloosterman sums. They are a necessary step in most proofs of subconvexity for Lfunctions, in a variety of equidistribution problems, in proofs of bounds and asymptotics for Maass forms, and in many applications of the circle method to problems in additive number theory. In attempting to generalize the Kuznetsov formula to study Fourier coefficents of SL(3) Maass forms, we encounter difficulties due to the lack of knowledge of the generalized Bessel functions that appear. The goal of this paper is to solve those difficulties for the spectral Kuznetsov formula.There are essentially four versions of the spectral Kuznetsov formula: The first, due to Xiaoqing Li, appears in [9, Thm 11.6.19] and uses an idea of Don Zagier with bi-K invariant test functions and spherical inversion. The second, due to Valentin Blomer, appears in [3] and uses the classical construction with the inner product of two Poincaré series. In [10], Dorian Goldfeld and Alex Kontorovich applied Lebedev-Whittaker inversion to Blomer's construction to obtain an arbitrary test function on the spectrum. Finally, in [7], the author worked out the first part of the integral transform on the geometric side of Li's version to obtain a slightly more usable formula -the main thrust there being an approximation to the geometric Kuznetsov formula.These four versions have been used with varying degrees of success. Li was able to obtain some results on spectral averages of the 1, 1 Fourier coefficients of cusp forms, weighted by their Whittaker functions [12]. Blomer has given a non-optimal version of the spectral large sieve inequalities and Lindelöf on average for the second moment of SL(3) L-functions [3]. Blomer, Raulf, and the author have given results on the distribution of the Fourier coefficients of SL(3) forms, including a Sato-Tate law on average [4].

Bilinear Forms withGL3Kloosterman Sums and the Spectral Large Sieve

“…For N large, say N ≫ X 2 1 +X 2 2 , Corollary 1.5 is optimized with H 1 = X 3/4 1 X 1/4 2 , H 2 = X 1/4 1 X 3/4 2 , and reduces to a bound that can be seen to be inferior to Corollary 1.3. On the other hand, if N ≪ min(X 2 1 , X 2 2 ), then the optimal bound occurs with Bu2] has developed Mellin-Barnes integral representations for the weight functions occuring on the Kloosterman sum side of the Bruggeman-Kuznetsov formula. Blomer and Buttcane [BB] have used this formulation, with additional ideas, to obtain a subconvexity result for GL 3 Maass forms in the spectral aspect.…”

“…One pleasant feature of this integral expression is that the variables are practically separated, and the kernel function is easily bounded uniformly in all parameters. For comparison, the formulas of Buttcane [Bu2,Theorem 2] also directly separate the variables and only require a 2-fold integral, which should in principle be more efficient. However the tradeoff is that the kernel function is not as easy to bound, requiring one to work on multiple scales.…”

Global decomposition of GL(3) Kloosterman sums and the spectral large sieve

“…. )dπ for a combined sum/integral over an orthonormal basis of spectral components of L 2 (SL 3 (Z)\H 3 ), which effectively runs over Hecke-Maaß cusp forms and Eisenstein series 1 (see [Go,Theorem 10.13.1] 1 The constant function and the maximal Eisenstein series twisted by the constant function have no non-zero Hecke eigenvalues and therefore will not occur in (1.6) below. or [Bu,Theorem 4] for details). For a compact subset Ω ⊆ ia * we denote analogously by Ω (.…”

Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions

“…In GLp2q, a couple of coincidences allow us to identify the oscillatory integrals with some well-studied special functions, see [Mo97], [I02]. However, such a phenomenon does not exist in GLp3q and there turn out to be many unexpected analytic difficulties, see Buttcane [Bu13,Bu16]. Furthermore, the complicated formulae for the oscillatory integrals make the Kuznetsov trace formula for GLp3q challenging to apply, see Blomer-Buttcane [BlBu20].…”

“…(2) Refine our investigation of the archimedean aspect of the problem, say along the line of the series of papers [Bu13,Bu16,BlBu20,Bu20,Bu21]. The main goal is to provide good estimations for the integral transforms.…”