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

Perret-Gentil, Corentin

doi:10.1090/tran/7333

Distribution questions for trace functions with values in cyclotomic integers and their reductions
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by Corentin Perret-Gentil PDF

Trans. Amer. Math. Soc. 371 (2019), 4585-4629

Abstract:

We consider $\ell$-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.

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