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Beschreibung:
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.