Torsion points on isogenous abelian varieties

verfasst von

Gabriel A. Dill

Abstract

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau and Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin-Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Artikel

Journal

Compositio mathematica

Band

158

Seiten

1020-1051

Anzahl der Seiten

32

ISSN

0010-437X

Publikationsdatum

20.07.2022

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie

Elektronische Version(en)

https://doi.org/10.48550/arXiv.2011.05815 (Zugang: Offen)
https://doi.org/10.1112/S0010437X22007400 (Zugang: Geschlossen)