On inverted Kloosterman sums over finite fields
On inverted Kloosterman sums over finite fields
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DOI：
10.1007/s00209-024-03457-0
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发表时间：
2023-01
影响因子：
0.8
通讯作者：
Xin Lin;D. Wan
中科院分区：
文献类型：
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作者：
Xin Lin;D. Wan
从复数的角度用Deligne定理和从进位的角度用Sperber定理可以很好地理解有限域上的经典变量Kloosterman和。本文研究了不定变量Kloosterman和的复数估计和进估计，解决了Katz（有限域应用1(3):395-398,1995）的一个问题。我们将给出两个复杂的估计。第一个是基于高斯和的基础。第二种估计更深入，依赖于Adolphson-Sperber， Denef-Loeser和Fu对于扭转指数和的上同调结果。这更深层次的结果假设characteristicpdoes不是divideCombining Dwork 'sp-adic理论,我们也确定exactp-adic 0和l函数相关的两极的估值toinvertedn-variable通过爱车caseAs我们将要看到的,theinvertedn-variable通过总和比classicaln-variable更复杂的通过和在所有方面,我们的理解是不完整,部分原因是霍奇数字现在主要是2而不是1。
The classicaln-variable Kloosterman sums over finite fields are well understood by Deligne’s theorem from complex point of view and by Sperber’s theorem fromp-adic point of view. In this paper, we study the complex andp-adic estimates ofinvertedn-variable Kloosterman sums, addressing a question of Katz (Finite Fields Appl 1(3):395–398, 1995). We shall give two complex estimates. The first one is elementary based on Gauss sums. The second estimate is deeper, depending on the cohomological results of Adolphson–Sperber, Denef–Loeser and Fu for twisted toric exponential sums. This deeper result assumes that the characteristicpdoes not divideCombining with Dwork’sp-adic theory, we also determine the exactp-adic valuations for zeros and poles of the L-function associated toinvertedn-variable Kloosterman sums in the caseAs we shall see, theinvertedn-variable Kloosterman sum is more complicated than the classicaln-variable Kloosterman sum in all aspects in the sense that our understanding is less complete, partly because the Hodge numbers are now mostly 2 instead of 1.