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 [3] in the case of ℤp. 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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Acknowledgements
I would like to thank Mark Shusterman, Julio Andrade, Jon Keating, and Brian Conrey for several helpful conversations and comments on this manuscript, as well as the anonymous referee for many helpful comments. This research was supported by NSF grant DMS-2101491.
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Dedicated to Peter Sarnak who, among other things, taught me how to pronounce ‘Kloosterman’
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Sawin, W. The size of wild Kloosterman sums in number fields and function fields. JAMA 151, 303–341 (2023). https://doi.org/10.1007/s11854-023-0325-9
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DOI: https://doi.org/10.1007/s11854-023-0325-9