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

2022/09/06 by Will Sawin, Sawin, Will
Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Historical Geopolitical and Social Dynamics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2209.02170

openalex publication_date 2022/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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