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ON THE DISTRIBUTION OF THE MAXIMUM OF CUBIC EXPONENTIAL SUMS

Part of: Finite fields and commutative rings (number-theoretic aspects) (Co)homology theory Exponential sums and character sums Limit theorems

Published online by Cambridge University Press:  27 September 2018

Youness Lamzouri
Youness Lamzouri*
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada (lamzouri@mathstat.yorku.ca)
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Abstract

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as ‘Birch sums’. Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of $\ell$-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist $x\in [1,p]$ and $a\in \mathbb{F}_{p}^{\times }$ such that $|\sum _{n\leqslant x}\exp (2\unicode[STIX]{x1D70B}i(n^{3}+an)/p)|\geqslant (2/\unicode[STIX]{x1D70B}+o(1))\sqrt{p}\log \log p$. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.

Keywords

Birch sumsExponential sumsSato–Tate distributionl-adic trace functionsRiemann hypothesis over finite fields

MSC classification

Primary: 11L03: Trigonometric and exponential sums, general 11T23: Exponential sums
Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 60F10: Large deviations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 4 , July 2020 , pp. 1259 - 1286
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Copyright
© Cambridge University Press 2018

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Footnotes

The author is partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

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