RESTRICTION OF SECTIONS FOR FAMILIES OF ABELIAN VARIETIES

Abstract. Given a family of Abelian varieties over a positive-dimensional base, we prove that for a sufficiently general curve in the base, every rational section of the family over the curve is contained in a unique rational section over the entire base. 1. Main results sec-intro The starting point for this article is the following theorem. Theorem 1.1. [GHMS05, Theorem 6.2] Let B be a smooth, quasi-projective, thm-GHMS complex variety, let A → B be an Abelian scheme over B (i.e. a family of Abelian varieties over B), and let pi: T → B be a torsor for A → B. Then pi is a trivial torsor if and only if for every curve C ⊂ B, the restriction TC → C is a trivial torsor for C ×B A → C. Stated more succinctly, the following map of étale cohomology groups is injectiv