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 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⊆P1k 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.