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Gaussian distribution of short sums of trace functions over finite fields

Published online by Cambridge University Press:  20 March 2017

CORENTIN PERRET–GENTIL
CORENTIN PERRET–GENTIL*
Affiliation:
ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland. e-mail: corentin.perretgentil@math.ethz.ch
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 385 - 422
Copyright
Copyright © Cambridge Philosophical Society 2017 

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