Dragos Ghioca, Thomas J. Tucker, and Michael E. Zieve:
The Mordell–Lang question for endomorphisms of semiabelian varieties,
J. Théor. Nombres Bordeaux 23 (2011), 645–666.

(The published version is available online.)

In this paper we reinterpret the Mordell–Lang conjecture as an assertion about translations on a semiabelian variety, and we analyze the analogous assertion for other classes of endomorphisms of semiabelian varieties. In more detail, Lang's generalization of the Mordell conjecture describes the intersection of certain subgroups and subvarieties of algebraic groups. Specifically, this conjecture addresses semiabelian varieties, namely, the connected algebraic groups G that are extensions of an abelian variety by an algebraic torus. The Mordell–Lang conjecture (now a theorem of Faltings and Vojta) asserts that if G is a semiabelian variety (over C), V is a closed subvariety of G, and T is a finitely generated subgroup of G(C), then V(C)∩T is the union of finitely many cosets of subgroups of T. We reinterpret the conclusion by viewing T as the collection of images of the origin under the (finitely-generated) semigroup of translations of G by elements of T. This suggests the following more general question:

Let G be a semiabelian variety, let f1,...,fr be commuting endomorphisms of G, let V be a closed subvariety of G, and let P be an element of G(C). Let E be the set of tuples (n1,...,nr) in Nr for which f1n1...frnr(P) lies in V(C). Is E the union of fintely many sets of the form u+(NrH) with u in Nr and H a subgroup of Zr?

The Faltings–Vojta theorem implies that the answer is "yes" if each fi is a translation (even if P isn't 0). We show that the answer is generally "no" if the fi's include both translations and algebraic group endomorphisms, but the answer is usually (but not always) "yes" if every fi is an algebraic group endomorphism.