Integral Points on Subvarieties of Semiabelian Varieties, Ii

This paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor, generalizing the corresponding result of Faltings for abelian varieties. Combined with the main theorem of the first part of this paper, this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety, minus a divisor. In addition, the last two sections generalize some standard results on closed subvarieties of semiabelian varieties to the context of closed subvarieties minus divisors. Recall that a semiabelian variety is a group variety A such that, after suitable base change, there exists an abelian variety A0 and an exact sequence (0.1) 0 → Gm → A ρ −→ A0 → 0 . (In this paper a variety is an integral separated scheme of finite type over a field. Since a group variety has a rational point, the base field is algebraically closed in the function field. In characteristic zero, this implies that the variety is geometrically integral.) Let k be a number field, and let S be a finite set of places of k containing all archimedean places. Let R be the ring of integers of k and let RS be the localization of R away from places in S . Let X be a quasi-projective variety over k . A model for X over RS is an integral scheme, surjective and quasi-projective over SpecRS , together with an isomorphism of the generic fiber over k with X . An integral point of X (or, loosely speaking, an integral point of X ) is an element of X (RS) . The first part [V 3] of this paper proved a finiteness statement (Theorem 0.3) for families of integral points on closed subvarieties X of a semiabelian variety A over k . This second and final part proves a similar result (Theorem 0.2) for certain open subvarieties of A . 1991 Mathematics Subject Classification. 11G10 (Primary); 11J25, 14G05, 14K15 (Secondary).