## 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 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 language | English (US) |
---|---|

Pages (from-to) | 303-341 |

Number of pages | 39 |

Journal | Journal d'Analyse Mathematique |

Volume | 151 |

Issue number | 1 |

DOIs | |

State | Published - Dec 2023 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- General Mathematics