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

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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.

Original languageEnglish (US)
Pages (from-to)303-341
Number of pages39
JournalJournal d'Analyse Mathematique
Issue number1
StatePublished - Dec 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • General Mathematics


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