Kloosterman paths and the shape of exponential sums

Emmanuel Kowalski; William F. Sawin

doi:10.1112/S0010437X16007351

Kloosterman paths and the shape of exponential sums

Emmanuel Kowalski, William F. Sawin

Mathematics

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums , as varies over and as 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.

Original language	English (US)
Pages (from-to)	1489-1516
Number of pages	28
Journal	Compositio Mathematica
Volume	152
Issue number	7
DOIs	https://doi.org/10.1112/S0010437X16007351
State	Published - Jul 1 2016

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Keywords

Kloosterman sheaves
Kloosterman sums
probability in Banach spaces
random Fourier series
Riemann Hypothesis over finite fields
short exponential sums

Access to Document

10.1112/S0010437X16007351

Cite this

@article{07c47435637a4eeaad36faa1eb8f325f,

title = "Kloosterman paths and the shape of exponential sums",

abstract = "We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums , as varies over and as 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 sheaves, Kloosterman sums, probability in Banach spaces, random Fourier series, Riemann Hypothesis over finite fields, short exponential sums",

author = "Emmanuel Kowalski and Sawin, {William F.}",

note = "Publisher Copyright: {\textcopyright} The Authors 2016.",

year = "2016",

month = jul,

day = "1",

doi = "10.1112/S0010437X16007351",

language = "English (US)",

volume = "152",

pages = "1489--1516",

journal = "Compositio Mathematica",

issn = "0010-437X",

publisher = "Cambridge University Press",

number = "7",

}

TY - JOUR

T1 - Kloosterman paths and the shape of exponential sums

AU - Kowalski, Emmanuel

AU - Sawin, William F.

N1 - Publisher Copyright: © The Authors 2016.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums , as varies over and as 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.

AB - We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums , as varies over and as 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.

KW - Kloosterman sheaves

KW - Kloosterman sums

KW - probability in Banach spaces

KW - random Fourier series

KW - Riemann Hypothesis over finite fields

KW - short exponential sums

UR - http://www.scopus.com/inward/record.url?scp=84963520703&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84963520703&partnerID=8YFLogxK

U2 - 10.1112/S0010437X16007351

DO - 10.1112/S0010437X16007351

M3 - Article

AN - SCOPUS:84963520703

SN - 0010-437X

VL - 152

SP - 1489

EP - 1516

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 7

ER -

Kloosterman paths and the shape of exponential sums

Abstract

All Science Journal Classification (ASJC) codes

Keywords

Access to Document

Other files and links

Fingerprint

Cite this