Model theory of difference fields with an additive character on the fixed field

TL;DR

This paper studies a model theory for difference fields with an additive character on the fixed field, establishing simplicity, characterizing amalgamation, and describing expansions with geometric elimination of imaginaries.

Contribution

It introduces the theory ACFA+ as a model companion, fully characterizes 3-amalgamation, and confirms the abelian nature of the Kim-Pillay group, advancing the understanding of difference fields with additive characters.

Findings

ACFA+ is a simple theory in characteristic 0.

Connected component of Kim-Pillay group is abelian.

Expansion of ACFA+ with geometric elimination of imaginaries is described.

Abstract

Following a research line proposed by Hrushovski in his work on pseudofinite fields with an additive character, we investigate the theory $\mathrm{ACFA}^{+}$ which is the model companion of the theory of difference fields with an additive character on the fixed field added as a continuous logic predicate. $\mathrm{ACFA}^{+}$ is the common theory (in characteristic $0$) of the algebraic closure of finite fields with the Frobenius automorphism and the standard character on the fixed field and turns out to be a simple theory. We fully characterise 3-amalgamation and deduce that the connected component of the Kim-Pillay group (for any completion of $\mathrm{ACFA}^{+}$) is abelian as conjectured by Hrushovski. Finally, we describe a natural expansion of $\mathrm{ACFA}^{+}$ in which geometric elimination of continuous logic imaginaries holds.