The Elementary Theory of the Frobenius Automorphisms

By Ehud Hrushovski. 2004

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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.

Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7be83f5d3ad93222fcd65f5dccdb54229724f570

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Combinatorial Lefschetz theorems beyond positivity (2018, background): “Venkatesh’s conjectures in the area [Ven17]. ◦Another possible area of interest and source of interesting deformations could be Hrushovski’s theory of Frobenius automorphisms and difference varieties [Hru04], but for this a cohomology theory remains to be developed. See also [Gia11] for recent devel”
Growth in finite simple groups of Lie type of bounded rank (2010, background): “Gσ) = Z(G). Second, if a σ-invariant connected closed subgroup of Gis normalised by Gσ then it is in fact normal in G. In this generality Theorem 6 depends on Hrushovski’s twisted Lang-Weil estimates [31]. In the proof of Theorem 4 this can be avoided (see Remark 71). Hence the constants in this the”
Periodic points of birational transformations on projective surfaces (2015, background): “t; x p > 2−(2/3)np−1. 6. The case λ1 > 1 The purpose of this section is to prove Theorem 1.2. 6.1. The case of ﬁnite ﬁelds. At ﬁrst, we recall the following theorem of Hrushovski. Theorem 6.1. ([20]) Let V be an irreducible aﬃne variety of degree d, dimension r over an algebraically closed ﬁel”
Expansion in finite simple groups of Lie type (2013, unknown): “rel subgroup. The nite group W:= N=T is known as the Weyl group of G, and is of order O(1). We then have the decomposition G = G w2W UTn wU w 14In principle, we could invoke the results of Hrushovski [35] here as was done in [24]; the model-theoretic arguments in that paper do not seem to readily gi”