Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves

Abstract

Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields k to the the specialization theorem for Néron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface S, for all vertical curves Sx of a fibration S → U ⊆ P 1 k with x from the complement of a sparse subset of |U |, the Mordell-Weil rank of an abelian scheme over S stays the same when restricted to Sx.