The Elementary Theory of the Frobenius Automorphisms

TL;DR

This paper develops a decision procedure for sentences in the language of Frobenius difference fields, generalizing classical theorems and exploring the structure of difference varieties with applications to finite groups and difference equations.

Contribution

It introduces a decision procedure for Frobenius difference fields and studies the structure of difference varieties of transformal dimension zero, extending classical algebraic results.

Findings

Decision procedure for Frobenius difference fields

Generalization of Cebotarev's density theorem and Weil's Riemann hypothesis

Applications to finite simple groups and difference equations

Abstract

A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending to infinity. In particular, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field. This generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves (both in qualitative versions), but hinges on a result going slightly beyond the latter. The setting for the proof is the geometry of difference varieties of transformal dimension zero; these generalize algebraic varieties, and are shown to have a rich structure, only partly explicated here. Some applications are given, in particular to finite simple groups, and to the Jacobi bound for difference equations.