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

verfasst von

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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Artikel

Journal

International Journal of Number Theory

Band

19

Seiten

1671-1680

Anzahl der Seiten

10

ISSN

1793-0421

Publikationsdatum

27.03.2023

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie

Elektronische Version(en)

https://doi.org/10.48550/arXiv.2301.12816 (Zugang: Offen)
https://doi.org/10.1142/S1793042123500811 (Zugang: Geschlossen)