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Transactions of the American Mathematical Society

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


Author: Corentin Perret-Gentil
Journal: Trans. Amer. Math. Soc. 371 (2019), 4585-4629
MSC (2010): Primary 11L05, 11T24, 11N64, 14F20, 60G50
DOI: https://doi.org/10.1090/tran/7333
Published electronically: August 9, 2018
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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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Corentin Perret-Gentil
Affiliation: Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland
Address at time of publication: Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec H3C 3J7, Canada
Email: corentin.perretgentil@gmail.com

DOI: https://doi.org/10.1090/tran/7333
Received by editor(s): October 27, 2016
Received by editor(s) in revised form: June 20, 2017
Published electronically: August 9, 2018
Additional Notes: This work was partially supported by DFG-SNF lead agency program grant 200021L_153647. The final corrections were made while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester, supported by the National Science Foundation under grant No. DMS-1440140. The results also appear in the author’s PhD thesis “Probabilistic aspects of short sums of trace functions over finite fields”.
Article copyright: © Copyright 2018 by Corentin Perret-Gentil