TY - JOUR

T1 - Unlikely intersections between isogeny orbits and curves

AU - Dill, Gabriel A.

N1 - Funding Information: Acknowledgments. I thank my advisor Philipp Habegger for suggesting this problem, for his continuous encouragement and for many helpful and interesting discussions. I thank Fabrizio Barroero, Philipp Habegger and Gaël Rémond for helpful comments on a preliminary version of this article. I thank Gregorio Baldi, whose article brought the conjecture of Buium and Poonen to my attention, and Fabrizio Barroero for pointing out the connection to Baldi’s article. I thank the anonymous referee for their helpful suggestions for improving the exposition. This work was partially supported by the Swiss National Science Foundation as part of the project “Diophantine Problems, o-Minimality, and Heights”, no. 200021 165525. Publisher Copyright: © 2021 European Mathematical Society.

PY - 2021

Y1 - 2021

N2 - Fix an abelian variety A0 and a non-isotrivial abelian scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of translates of a fixed finite-rank subgroup of A0, also defined over the algebraic numbers, by abelian subvarieties of A0 of codimension at least k under all isogenies between A0 and some fiber of the abelian scheme. We characterize the curves inside the abelian scheme which are defined over the algebraic numbers, dominate the base curve and potentially intersect this set in infinitely many points. Our proof follows the Pila-Zannier strategy.

AB - Fix an abelian variety A0 and a non-isotrivial abelian scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of translates of a fixed finite-rank subgroup of A0, also defined over the algebraic numbers, by abelian subvarieties of A0 of codimension at least k under all isogenies between A0 and some fiber of the abelian scheme. We characterize the curves inside the abelian scheme which are defined over the algebraic numbers, dominate the base curve and potentially intersect this set in infinitely many points. Our proof follows the Pila-Zannier strategy.

KW - Abelian scheme

KW - Andre-Pink-Zannier conjecture

KW - Isogeny

KW - Unlikely intersections

UR - http://www.scopus.com/inward/record.url?scp=85108441815&partnerID=8YFLogxK

U2 - 10.4171/JEMS/1057

DO - 10.4171/JEMS/1057

M3 - Article

AN - SCOPUS:85108441815

VL - 23

SP - 2405

EP - 2438

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 7

ER -