Gaussian distribution of short sums of trace functions over finite fields
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Date
2017-11
Publication Type
Journal Article
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Abstract
We show that under certain general conditions, short sums of ℓ-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalising results of Erdős–Davenport, Mak–Zaharescu and Lamzouri. In particular, this applies to exponential sums arising from Fourier transforms such as Kloosterman sums or Birch sums, as we can deduce from the works of Katz. By approximating the moments of traces of random matrices in monodromy groups, a quantitative version can be given as in Lamzouri's article, exhibiting a different phenomenon than the averaging from the central limit theorem.
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published
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163 (3)
Pages / Article No.
385 - 422
Publisher
Cambridge University Press
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It was possible to publish this article open access thanks to a Swiss National Licence with the publisher.
Funding
153647 - Geometric and Analytic Number Theory (SNF)