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.

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We show that integral monodromy groups of Kloosterman ℓ-adic sheaves of rank n ě 2 on Gm{Fq are as large as possible when the characteristic ℓ is large enough, depending only on the rank. This variant of Katz's results over C was known by works of Gabber, Larsen, Nori and Hall under restrictions such as ℓ large enough depending on charpFqq with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz's ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman-Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.

We consider ℓ-adic trace functions over finite fields taking values in cyclotomic integers, such as characters and exponential sums. Through ideas of Deligne and Katz, we explore probabilistic properties of the reductions modulo a prime ideal, exploiting especially the determination of their integral monodromy groups. In particular, this gives a generalization of a result of Lamzouri-Zaharescu on the distribution of short sums of the Legendre symbol reduced modulo an integer to all multiplicative characters and to hyper-Kloosterman sums.