(Both the published version and the arXiv version are 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,...,f_r 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,...,n_r) in N^r for which f₁^n₁...f_r^n_r(P) lies in V(C). Is E the union of fintely many sets of the form u+(N^r ∩ H) with u in N^r and H a subgroup of Z^r?

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

Michael Zieve:  home page   publication list