Parametrization of Kloosterman sets and SL₃-Kloosterman sums

2020/01/07 by Kıral, Eren Mehmet, Nakasuji, Maki
#11F72 (secondary) #11L05 (primary) #FOS: Mathematics #Number Theory (math.NT) #and 11F22

paper · doi:10.48550/arxiv.2001.01936

Abstract

We stratify the SL₃ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the SL₃ 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 Γ₀(N)⊆ SL₃(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.

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