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

TL;DR

This paper investigates the distribution of trace functions over finite fields with values in cyclotomic integers, analyzing their reductions modulo primes and generalizing known results on character sums and exponential sums.

Contribution

It introduces a framework for studying the distribution of trace functions in cyclotomic integers and extends existing results to a broader class of sums using monodromy group analysis.

Findings

Generalization of Lamzouri-Zaharescu result to all multiplicative characters

Distribution properties of hyper-Kloosterman sums modulo primes

Connection between monodromy groups and sum distributions

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.