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

authored by

Timo Keller

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 ⊆Pk1 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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

International Journal of Number Theory

Volume

19

Pages

1671-1680

No. of pages

10

ISSN

1793-0421

Publication date

27.03.2023

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory

Electronic version(s)

https://doi.org/10.48550/arXiv.2301.12816 (Access: Open)
https://doi.org/10.1142/S1793042123500811 (Access: Closed)