Algebraically Closed Fields with a Generic Multiplicative Character

Abstract

We study the model theory of the $2$-sorted structure $(\mathbb{F},\mathbb{C};\chi )$, where $\mathbb{F}$ is an algebraic closure of a finite field of characteristic $p$, $\mathbb{C}$ is the field of complex numbers and $\chi :\mathbb{F}\to \mathbb{C}$ is an injective, multiplication preserving map. We obtain an axiomatization ${\mathrm{ACFC}}_p$ of $\mathrm{Th}(\mathbb{F},\mathbb{C};\chi )$ in a suitable language $L$, classify the models of ${\mathrm{ACFC}}_p$ up to isomorphism, prove a modified model companion result, give various descriptions of definable sets inside a model of ${\mathrm{ACFC}}_p$, and deduce that ${\mathrm{ACFC}}_p$ is $\omega$-stable and has definability of Morley rank in families.

36 pages