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Kloosterman paths and the shape of exponential sums

Part of:Stochastic processes(Co)homology theoryFinite fields and commutative rings (number-theoretic aspects)Limit theoremsExponential sums and character sums

Published online by Cambridge University Press:  15 April 2016

Emmanuel Kowalski and
William F. Sawin
Emmanuel Kowalski
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, CH-8092 Zürich, Switzerland email kowalski@math.ethz.ch
William F. Sawin
Affiliation:
Princeton University, Fine Hall, Washington Road, NJ, USA email wsawin@math.princeton.edu
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Abstract

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We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums $\text{Kl}_{p}(a)$ , as $a$ varies over $\mathbf{F}_{p}^{\times }$ and as $p$ tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

Keywords

Kloosterman sumsKloosterman sheavesRiemann Hypothesis over finite fieldsrandom Fourier seriesshort exponential sumsprobability in Banach spaces

MSC classification

Primary:11T23: Exponential sums11L05: Gauss and Kloosterman sums; generalizations14F20: Étale and other Grothendieck topologies and (co)homologies60F17: Functional limit theorems; invariance principles60G17: Sample path properties
Secondary:60G50: Sums of independent random variables; random walks

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 7 , July 2016, pp. 1489 - 1516
Copyright
© The Authors 2016 

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