Fourth power moment of twisted Kloosterman sum and Hurwitz class numbers

TL;DR

This paper derives an explicit formula for the fourth power moment of twisted Kloosterman sums, linking it to Hurwitz class numbers, and analyzes its asymptotic behavior using advanced modular form theories.

Contribution

It introduces a new explicit formula connecting twisted Kloosterman sums with Hurwitz class numbers and studies their asymptotic behavior using harmonic Maass forms.

Findings

Explicit formula for the fourth power moment in terms of Hurwitz class numbers

Asymptotic formulas for weighted sums of Hurwitz class numbers

Asymptotic behavior of the fourth power moment of twisted Kloosterman sums

Abstract

In this paper, we investigate the fourth power moment of twisted Kloosterman sum and its relationship with Hurwitz class number. We derive an explicit formula expressing this moment in terms of weighted sums involving Hurwitz class numbers. Our approach involves analyzing point counting formulas associated with the resolution of certain Calabi-Yau threefold. Furthermore, we study the asymptotic behaviour of weighted sums of Hurwitz class numbers that appear in the moment formula. To derive these asymptotic formulas, we employ the theory of harmonic Maass forms, mock modular forms and holomorphic projections. As an application of these asymptotic results, we obtain the asymptotic formula for the fourth power moment of twisted Kloosterman sums.