Correction to ?On the monodromy groups attached to certain families of exponential sums?

AbstractWe 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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Exponential Sums Over Finite Fields and the Large Sieve

International Mathematics Research Notices ◽

10.1093/imrn/rny202 ◽

2018 ◽

Vol 2020 (20) ◽

pp. 7139-7174 ◽

Cited By ~ 1

Author(s):

Corentin Perret-Gentil

Keyword(s):

Finite Fields ◽

Exponential Sums ◽

Kloosterman Sums ◽

Large Sieve ◽

Density Estimates ◽

Zero Density ◽

Monodromy Groups ◽

Cyclotomic Integers

Abstract By using a variant of the large sieve for Frobenius in compatible systems developed in [24] and [27], we obtain zero-density estimates for arguments of $\ell $-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.

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EXPONENTIAL SUMS

A-to-Z Guide to Thermodynamics Heat and Mass Transfer and Fluids Engineering ◽

10.1615/atoz.e.expsum ◽

2006 ◽

Vol e ◽

Author(s):

H.G Khajah

Keyword(s):

Exponential Sums

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A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

Advances in Difference Equations ◽

10.1186/s13662-021-03353-5 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Wenpeng Zhang ◽

Xingxing Lv

Keyword(s):

Exponential Sums ◽

Exponential Sum ◽

Gauss Sums ◽

Power Mean ◽

Hybrid Power ◽

Analytic Methods ◽

Hybrid Power Mean

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

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Independence of algebraic monodromy groups in compatible systems

Selecta Mathematica ◽

10.1007/s00029-021-00657-y ◽

2021 ◽

Vol 27 (4) ◽

Author(s):

Federico Amadio Guidi

Keyword(s):

Galois Representations ◽

Global Function ◽

Irreducible Decomposition ◽

Global Function Fields ◽

Normal Varieties ◽

Independence Result ◽

Monodromy Groups ◽

Independence Results ◽

Lisse Sheaves ◽

General Method

AbstractIn this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classes of automorphic forms, we prove the existence of a Lie-irreducible decomposition. From this we deduce an independence result. We conclude with the case of compatible systems of Galois representations over global function fields, for which we prove the existence of a Lie-irreducible decomposition, and we deduce an independence result. From this we also deduce an independence result for compatible systems of lisse sheaves on normal varieties over finite fields.

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On Waring’s problem: Three cubes and a sixth power

Nagoya Mathematical Journal ◽

10.1017/s0027763000007893 ◽

2001 ◽

Vol 163 ◽

pp. 13-53 ◽

Cited By ~ 4

Author(s):

Jörg Brüdern ◽

Trevor D. Wooley

Keyword(s):

Exponential Sums ◽

Large Integer ◽

Natural Numbers ◽

Waring’S Problem ◽

Smooth Numbers ◽

Waring's Problem ◽

Order Of Magnitude ◽

Almost All ◽

Hardy Littlewood Method

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

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Construction of systems of polynomial equations with exact p-Divisibility via the covering method

Journal of Algebra and Its Applications ◽

10.1142/s0219498814500133 ◽

2014 ◽

Vol 13 (06) ◽

pp. 1450013 ◽

Cited By ~ 3

Author(s):

Francis N. Castro ◽

Ivelisse M. Rubio

Keyword(s):

Exponential Sums ◽

Polynomial Equations ◽

Prime Field ◽

Elementary Method ◽

Covering Method ◽

Systems Of Polynomial Equations

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

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