Distribution questions for trace functions with values in cyclotomic integers and their reductions

Perret-Gentil, Corentin

doi:10.1090/tran/7333

Cited by 3 publications

(2 citation statements)

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“…The Birch sums (1.2) are examples of ℓ-adic trace functions over finite fields. These trace functions have been extensively studied in a series of recent works by Fouvry, Kowalski, and Michel [4], [5], [6], Fouvry, Kowalski, Michel, Raju, Rivat, and Soundararajan [7], Kowalski and Sawin [11], [12], and Perret-Gentil [16], [17]. Our results can be generalized to other types of trace functions that are attached to certain coherent families of ℓ-adic sheaves (in the sense given by Perret-Gentil [16]), if their short sums satisfy a bound similar to (1.8).…”

Section: Introductionmentioning

confidence: 99%

On the Distribution of the Maximum of Cubic Exponential Sums

Lamzouri¹

2018

J. Inst. Math. Jussieu

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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 ℓ-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 ∈ [1, p] and a ∈ F × p such that | n≤x exp(2πi(n 3 + an)/p)| ≥ (2/π + o(1)) √ p log log p. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.

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Section: Introductionmentioning

confidence: 99%

On the Distribution of the Maximum of Cubic Exponential Sums

Lamzouri¹

2018

J. Inst. Math. Jussieu

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“…In [PG16a], we took the point of view of studying these as (normalized) algebraic integers, in the sense that they all take values in the localization Zrζ 4p s q 1{2 (by the evaluation of quadratic Gauss sums).…”

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confidence: 99%

Exponential sums over finite fields and the large sieve

Perret-Gentil

2017

Preprint

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By using a variant of the large sieve for Frobenius in compatible systems developed in [Kow06a] and [Kow08], we obtain zerodensity estimates for arguments of ℓ-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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Traces of powers of matrices over finite fields

Gorodetsky¹,

Rodgers²

2021

Trans. Amer. Math. Soc.

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Let g be a random matrix distributed according to uniform probability measure on the finite general linear group GLn(Fq). We show that Tr(g k ) equidistributes on Fq as n → ∞ as long as log k = o(n 2 ) and that this range is sharp. We also show that nontrivial linear combinations of Tr(g 1 ), . . . , Tr(g k ) equidistribute as long as log k = o(n) and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for k ≤ cqn, where cq depends on q, due to work of the first-named author and Rodgers.We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of Tr(g k ) we end up showing that certain explicit character sums modulo T k+1 exhibit cancellation when averaged over monic polynomials of degree n in Fq[T ] as long as log k = o(n 2 ). This goes far beyond the classical range log k = o(n) due to Montgomery and Vaughan. To study these sums we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.

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Distribution questions for trace functions with values in cyclotomic integers and their reductions

Cited by 3 publications

References 32 publications

On the Distribution of the Maximum of Cubic Exponential Sums

On the Distribution of the Maximum of Cubic Exponential Sums

Exponential sums over finite fields and the large sieve

Traces of powers of matrices over finite fields

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