On sums of $\mathit{SL}(3,\mathbb{Z})$ Kloosterman sums

Abstract

We show that sums of the SL(3, Z) long element Kloosterman sum against a smooth weight function have cancellation due to the variation in argument of the Kloosterman sums, when each modulus is at least the square root of the other. Our main tool is Li's generalization of the Kuznetsov formula on SL(3, R), which has to date been prohibitively difficult to apply. We first obtain analytic expressions for the weight functions on the Kloosterman sum side by converting them to Mellin-Barnes integral form. This allows us to relax the conditions on the test function and to produce a partial inversion formula suitable for studying sums of the long-element SL(3, Z) Kloosterman sums.

Bilinear Forms withGL3Kloosterman Sums and the Spectral Large Sieve

“…First we need an individual "Weil-type" bound. This estimate was proved by Stevens [S] but without explicit dependence on the m i and n i , which was subsequently investigated by Buttcane [Bu1,Theorem 4].…”

The arithmetic Kuznetsov formula on GL(3), I: The Whittaker case

“…In section 9, we will use Wallach's theorem for GLp3q (see section 6.10) along with bounds on the Whittaker and Bessel functions (see section 7), and the uniqueness of spherical functions (see section 6.11) to show that nice functions of Y can be expanded into (sums/integrals of) the GLp3q Bessel functions. Along the way, we show the generalization of [14,Lemma 3], the Fourier transform of the spherical function, to the weight one case, but we eventually arrive at:…”

“…The Fourier transform of a spherical function. Here we prove Proposition 29; the proof follows that of[14, Lemma 3]. By sending x Þ Ñ v´´x ι v´´and using the K-invariance…”

“…We will not require in-depth knowledge about the sum itself, beyond the square-root cancellation bound (originally due to Stevens [44], but later made global with explicit dependence on the indices in [14], and strengthened in [6, (2.10)]):…”

Higher weight on GL(3). I: The Eisenstein series

“…whose construction is explained in [4,Section 5]. In case n 2 = 0, we conjugate by w l and apply the dual representation D d , which reverses the rows and columns, and hence the roles of u 1 and u 2 .…”