Pseudofinite fields with additive and multiplicative character

TL;DR

This paper develops a logical framework for pseudofinite fields equipped with generic additive and multiplicative characters, establishing their asymptotic behavior, quantifier elimination, and simplicity within continuous logic.

Contribution

It introduces the theory $ ext{PF}^{+, imes}$ for pseudofinite fields with additive and multiplicative characters, proving asymptotic equivalence, quantifier elimination, and simplicity.

Findings

The theory $ ext{PF}^{+, imes}$ models the asymptotic behavior of finite fields with generic characters.

Quantifier elimination is achieved in a natural expansion of the language.

The theory is shown to be simple.

Abstract

We introduce the theory $\mathrm{PF}^{+,\times}$ of pseudofinite fields with generic additive and multiplicative character added as continuous logic predicates. Using the Weil bounds on character sums over finite fields as well as the Erd\H{o}s-Tur\`an-Koksma inequality we show that it is the asymptotic theory (in characteristic $0$) of finite fields with (sufficiently generic) additive and multiplicative character. Moreover, we establish quantifier elimination in a natural definitional expansion of the language and deduce that integration by the Chatzidakis-van den Dries-Macintyre counting measure is uniformly definable in the parameters. Finally, we show that $\mathrm{PF}^{+,\times}$ is a simple theory.