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

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 151; no. 1; pp. 303 - 341
• Main Author: Sawin, Will
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01.12.2023; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Analysis; Dirichlet problem; Dynamical Systems and Ergodic Theory; Functional Analysis; Lower bounds; Mathematics; Mathematics and Statistics; Number theory; Partial Differential Equations; Upper bounds
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-023-0325-9

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 [3] in the case of 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.