The size of wild Kloosterman sums in number fields and function fields

TL;DR

This paper investigates p-adic hyper-Kloosterman sums over various local rings, providing sharp bounds on their size and implications for number theory, especially regarding sums and L-functions in function fields.

Contribution

It extends bounds for hyper-Kloosterman sums to general p-adic and equal characteristic rings, with new stationary phase estimates and applications to function field number theory.

Findings

Sharp bounds for hyper-Kloosterman sums when k is not divisible by p

Bounds when k is divisible by p, showing sharpness of upper bounds

Applications demonstrating lack of square-root cancellation in certain sums and L-functions

Abstract

We study p-adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter k that recovers the classical Kloosterman sum when k=2, over general p-adic rings and even equal characteristic local rings. These can be evaluated by a simple stationary phase estimate when k is not divisible by p, giving an essentially sharp bound for their size. We give a more complicated stationary phase estimate to evaluate them in the case when k is divisible by p. This gives both an upper bound and a lower bound showing the upper bound is essentially sharp. This generalizes previously known bounds of Cochrane, Liu, and Zhen in the case of $\mathbb Z_p$. The lower bounds in the equal characteristic case have two applications to function field number theory, showing that certain short interval sums and certain moments of Dirichlet L-functions do not, as one might hope, admit square-root cancellation.