The Elementary Theory of the Frobenius Automorphisms

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.

v2 - local improvements based on comments of many people, especially during the CIRM workshop in Luminy in March 2013

Cited by in corpus (36)

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• Periodic points of birational maps on projective surfaces
• Combinatorial Lefschetz theorems beyond positivity
• Some properties of analytic difference fields
• On the Schmidt and analytic ranks for trilinear forms
• Existentially closed fields with G-derivations
• The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)
• A twisted theorem of Chebotarev
• Jacobi's bound and normal forms computations. A historical survey
• Difference Nullstellensatz in the case of finite group
• Expansion in finite simple groups of Lie type
• A Chevalley theorem for difference equations
• Finiteness properties of pseudo-hyperbolic varieties
• Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb{P}}^N$ over function fields
• Théorèmes d'équidistribution pour les systèmes dynamiques d'origine arithmétique
• Suzuki groups as expanders
• Difference modules and difference cohomology
• Projective Crystalline Representations of Étale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow
• On degrees of birational mappings
• A topos-theoretic view of difference algebra
• Geometry and arithmetic of verbal dynamical systems on simple groups
• The (non-uniform) Hrushovski-Lang-Weil estimates
• Algebraic entropy for smooth projective varieties
• Model theory of finite and pseudofinite groups
• A differential Chevalley theorem
• Quantifier Elimination for Valued Fields Equipped with an Automorphism
• Strongly dense free subgroups of semisimple algebraic groups
• Relative Rank and Regularization
• Multiplicity in difference geometry
• No iterated identities satisfied by all finite groups
• On the ascending chain condition for mixed difference ideals
• Grouplike minimal sets in ACFA and in T_A
• Fields with automorphism and valuation
• Uniform Definability and Undecidability in Classes of Structures
• An Approximate AKE Principle for Metric Valued Fields
• Decidability of the theory of addition and the Frobenius map in rings of rational functions