Submission date: 11 December 2017

Abstract:

The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group Gamma equipped with an arbitrary endomorphism F. This is applied to the isotrivial positive characteristic Mordell-Lang context where F is the Frobenius action on a commutative algebraic group G over a finite field, and Gamma is a finitely generated F-invariant subgroup of G. It is shown that the F-subsets of Gamma introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X is a closed subvariety then X intersect Gamma is F-automatic. Derksen's notion of a k-normal subset of the natural numbers is also here extended to the above abstract setting, and shown under certain hypotheses to coincide with the F-sets. In particular, the X intersect Gamma appearing in the Mordell-Lang problem are F-normal. When specialised to the case of G a multiplicative torus and Gamma cyclic, this is precisely Derksen's Skolem-Mahler-Lech theorem.

Mathematics Subject Classification: 11G25, 68Q45

Keywords and phrases: automatic sets, F-sets, Mordell-Lang, Skolem-Mahler-Lech

Full text arXiv 1712.03800: pdf, ps.