TY - JOUR

T1 - Torsion points on isogenous abelian varieties

AU - Dill, Gabriel A.

N1 - Funding Information: This paper has grown out of a section of my PhD thesis. I thank my PhD advisor Philipp Habegger for his constant support and for many helpful and interesting discussions. I thank Philipp Habegger and Gaël Rémond for helpful comments on the thesis. I thank Fabrizio Barroero and Gaël Rémond for useful remarks on a preliminary version of this paper. I thank Francesco Campagna, Davide Lombardo, César Martínez, David Masser, Jonathan Pila, and Harry Schmidt for useful conversations and correspondence. I thank the referee for their helpful suggestions and in particular for improving the exponent in Lemma from to , which also improved the upper bounds in Theorems and . When I had the initial idea for this paper, I was supported by the Swiss National Science Foundation as part of the project ‘Diophantine Problems, o-Minimality, and Heights’, no. 200021_165525. I completed it while supported by the Early Postdoc.Mobility grant no. P2BSP2_195703 of the Swiss National Science Foundation. I thank the Mathematical Institute of the University of Oxford and my host there, Jonathan Pila, for hosting me as a visitor for the duration of this grant.

PY - 2022/7/20

Y1 - 2022/7/20

N2 - 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.

AB - 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.

KW - abelian scheme

KW - effectivity

KW - homothety

KW - isogeny

KW - Manin-Mumford

KW - uniformity

UR - http://www.scopus.com/inward/record.url?scp=85134946747&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2011.05815

DO - 10.48550/arXiv.2011.05815

M3 - Article

AN - SCOPUS:85134946747

VL - 158

SP - 1020

EP - 1051

JO - Compositio mathematica

JF - Compositio mathematica

SN - 0010-437X

IS - 5

ER -