Supercharacters, elliptic curves, and the sixth moment of Kloosterman sums

TL;DR

This paper links the sixth moment of Kloosterman sums to elliptic curves, providing an elementary proof that these sums are bounded by a constant times p^{2/3} for certain u.

Contribution

It introduces a novel connection between Kloosterman sums and elliptic curves, offering a new elementary proof of their bounds.

Findings

Bound on Kloosterman sums: O(p^{2/3}) for u with p mid u

Connection established between sixth moment of sums and elliptic curves

Elementary proof technique for sum bounds

Abstract

We connect the sixth power moment of Kloosterman sums to elliptic curves. This yields an elementary proof that $K_u$ with $p\nmid u$ are $O(p^{2/3})$.