{"title":"The size of wild Kloosterman sums in number fields and function fields","authors":"","doi":"10.1007/s11854-023-0325-9","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We study <em>p</em>-adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter <em>k</em> that recovers the classical Kloosterman sum when <em>k</em> = 2, over general <em>p</em>-adic rings and even equal characteristic local rings. These can be evaluated by a simple stationary phase estimate when <em>k</em> is not divisible by <em>p</em>, giving an essentially sharp bound for their size. We give a more complicated stationary phase estimate to evaluate them in the case when <em>k</em> is divisible by <em>p</em>. 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<sub><em>p</em></sub>. 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 <em>L</em>-functions do not, as one might hope, admit square-root cancellation.</p>","PeriodicalId":502135,"journal":{"name":"Journal d'Analyse Mathématique","volume":"20 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal d'Analyse Mathématique","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11854-023-0325-9","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.
摘要 我们研究的是 p-adic 超克罗斯特曼和，它是带有参数 k 的克罗斯特曼和的广义化，当 k = 2 时，可以在一般 p-adic 环和偶数等特征局部环上恢复经典的克罗斯特曼和。当 k 不能被 p 整除时，可以通过简单的静止相位估计来评估这些和，从而为它们的大小给出一个基本的锐界。在 k 能被 p 整除的情况下，我们给出了一个更复杂的静态相位估计来评估它们。这同时给出了一个上界和一个下界，表明上界基本上是尖锐的。这概括了之前已知的ℤp 情况下的边界 [3]。等特征情况下的下界在函数场数理论中有两个应用，表明某些短区间和以及某些迪里夏特 L 函数矩并不像人们所希望的那样，允许平方根取消。
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