Gaussian distribution of short sums of trace functions over finite fields

Abstract

We 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, generalizing 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.

On the Distribution of the Maximum of Cubic Exponential Sums

“…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). The precise definition of a coherent family is technical (see [16]), but roughly speaking, these are sheaves over F p for which the "conductor" is bounded independently of p, the arithmetic and geometric monodromy groups are equal, of fixed type and large, and the sheaves formed by additive shifts are "independent". As an example, Theorem 1.1 can be generalized for the partial sums of the exponential sum…”

“…To this end, we shall use the recent work of Perret-Gentil [16] which relies on deep tools from algebraic geometry, in order to investigate the moments of sums of the form…”

Distribution of short sums of classical Kloosterman sums of prime powers moduli

“…Remark 4.3-Corentin Perret-Gentil mentioned that this result is hidden in [PG17] in a more theoretical language.…”

“…where as usual I q + x stands for the translate of I q by x for any x in F q . Given a sequence t q of -adic trace functions over F q and a sequence I q of subsets of F q , C. Perret-Gentil got interested in [PG17] in the distribution as q and |I q | tend to infinity of the sequence of complex-valued random variables S(t q , I q ; * ) and proved a deep general result under very natural conditions. Let us mention that his general result is not only a generalization but also an improvement over previous works such as [DE52], [MZ11], [Lam13b] and [Mic98].…”

“…Remark 1.3-This theorem is the analogue of Theorem 1.1. The condition (1.1) is new and comes from the context of finite rings in this work instead of finite fields in [PG17] whereas the condition (1.2) is exactly the same and is inherent to the method of proof itself namely the method of moments. Note that the condition (1.1) requires that |I p n | < p holds, which is automatically satisfied by (1.2).…”