F -structures and integral points on semiabelian varieties over finite fields

Abstract

Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination result for certain modules over finite simple extensions of the integers given together with predicates for orbits of the distinguished generator of the ring.Remark 2.2. For fixed R = Z[F ], the following observations are immediate consequences of the definitions:• The class of F -spaces is closed under taking products and passing to quotients by F -subspaces.

The isotrivial case in the Mordell-Lang Theorem

“…In Section 3 we will prove our main theorem for semiabelian varieties, while in Section 4 we will show how our statement for G g a can be proved along similar lines as the result in [8]. We note that our Mordell-Lang statement for the additive group scheme is actually a Mordell-Lang statement for Drinfeld modules defined over finite fields.…”

“…The classical example of a Frobenius ring associated to a semiabelian variety G defined over the finite field F q is Z[F ], where F ∈ End(G) is the endomorphism of G induced by the Frobenius map on F q . This Frobenius ring is discussed in [8] and [9]. Also, …”

“…In [1], Abramovich and Voloch conjecture a function-field version of the Mordell-Lang statement in positive characteristic by formulating a condition which has to be satisfied by the subvariety X, if X has a Zariski dense intersection with a finitely generated subgroup Γ ⊂ G. The conjecture is proved, using modeltheoretic techniques, by Hrushovski in [5]. However, [5] does not offer a description of the intersection of X with Γ. Moosa and Scanlon provide in [8] and [9] a first answer toward the last question in the isotrivial case, under an extra assumption on the subgroup Γ. In the present paper we are able to remove their assumption.…”

“…Let X be a subvariety of G defined over K (in this paper, all subvarieties will be closed). In [8] and [9], Moosa and Scanlon discussed the intersection of the Kpoints of X with a finitely generated Z[F ]-submodule Γ of G(K). They proved that the intersection is a finite union of F -sets in Γ (see Definition 2.4).…”

Dynamical Mordell-Lang conjecture in positive characteristic

“…In Section 2 we discuss the classical Mordell-Lang problem for algebraic tori in characteristic p, by stating the results of Moosa-Scanlon [MS04] and of Derksen-Masser [DM12] and then explaining its connections to Conjecture 1.1. In Section 3 we introduce the so-called p-arithmetic sequences, i.e., sets of the form (2) and discuss properties of these sequences including in the larger context of linear recurrence sequences.…”