Parametrization of Kloosterman sets and ${\mathrm{SL}}_3$-Kloosterman sums

Abstract

We stratify the ${\mathrm{SL}}_3$ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the ${\mathrm{SL}}_3$ long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup ${\Gamma }_0(N)\subseteq {\mathrm{SL}}_3(\mathbb{Z})$. Another application is a new explicit formula, expressing the triple divisor sum function in terms of a double Dirichlet series of exponential sums, generalizing Ramanujan's formula.

39 pages

References in corpus (1)

• On the GL(3) Kuznetsov Formula with applications to Symmetry Types of families of $L$-functions