Twisted Lang-Weil estimates and definable quotients

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Abstract

By a celebrated result of Ax from the 60s, everything non-trivial that can be said about ’large’ finite fields follows from the Lang-Weil estimates. This may be made precise with the help of so-called pseudofinite fields. A twisted version of the Lang-Weil estimates, due to Hrushovski, leads to an analogous result on the q-Frobenius automorphism, where q is a ’large’ prime power, ultimately showing that the non-standard Frobenius automorphism satisfies a Nullstellensatz for difference polynomials. In the talk, we will first give an overview of these results, and we will then present a classification of definable quotients in algebraically closed fields endowed with a non-standard Frobenius automorphism, also in case a non-trivial valuation is added to the structure.

Speakers from the University of Münster

Hils, Martin

Professorship for Mathematical Logic (Prof. Hils)

Projects the talk is about

Geometric and Combinatorial Configurations in Model Theory (GeoMod) Duration: 01/01/2020 - 31/12/2024 Funded by: DFG - Individual Grants Programme Type of project: Individual project

EXC 2044 - A2: Groups, model theory and sets Duration: 01/01/2019 - 31/12/2025 | 1st Funding period Funded by: DFG - Cluster of Excellence Type of project: Subproject in DFG-joint project hosted at University of Münster

Publications referred to in the talk

Un principe d'Ax-Kochen-Ershov imaginaire Hils Martin, Rideau-Kikuchi Silvain (2026) In: Journal of the European Mathematical Society, 28(3), 1009-1080. doi:10.4171/JEMS/1492 Research article (journal) | Peer reviewed | Published