THOMAS SCANLON AND JOSÉ FELIPE VOLOCH

DIFFERENCE SUBGROUPS OF COMMUTATIVE ALGEBRAIC GROUPS OVER FINITE FIELDS

Abstract. The work of Chatzidakis and Hrushovski on the model theory of difference fields in characteristic zero showed that groups defined by difference equations have a very restricted structure. For instance, if G is a semi-abelian variety over a difference field of characteristic zero and Γ ⊂ G is a subgroup of “modular type”, then for any subvariety X ⊂ G, X ∩ Γ is a finite union of cosets of subgroups of Γ. Using such facts once can resolve diophantine questions about special subgroups of G (for instance, the torsion subgroup). Recent work of Chatzidakis, Hrushovski and Peterzil [CHP] extends the class of difference fields for which this sort of result is known to positive characteristic. In this note, we analyze the subgroups of the torsion points of simple commutative algebraic groups over finite fields that can be constructed by such difference equations. Our results are reasonably complete modulo some well-known conjectures in Number Theory. In one case, we need the p-adic version of the four exponentials conjecture and in another we need a version of Artin’s conjecture on primitive roots.

We recover part of a theorem of Boxall on the intersection of varieties with the group of m-power torsion points, but in general this theorem does not follow from the model-theoretic analysis, because there may be no field automorphism σ so that the m-power torsion group is contained in a modular group definable with σ. On the other hand, some of the groups defined by modular difference equations are much larger than the group of m-power torsion points, so our results are stronger in another direction. In some ways, the model theoretic approach extends the approach of Bogomolov and the original one of Lang.