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

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 $S_x$ of a fibration $S\to U\subseteq {\mathbf{P}}_k^1$ with $x$ from the complement of a sparse subset of $\mathrm{\mid }U\mathrm{\mid }$, the Mordell-Weil rank of an abelian scheme over $S$ stays the same when restricted to $S_x$.