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arXiv · 2512.10583

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

Stefan Marian Ludwig
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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.

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Stefan Marian Ludwig. 2025-12-11. Model theory of difference fields with an additive character on the fixed field. https://arxiv.org/abs/2512.10583

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