Unlikely intersections between isogeny orbits and curves
Abstract
Fix an abelian variety $A_0$ and a nonisotrivial abelian scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of translates of a fixed finiterank subgroup of $A_0$, also defined over the algebraic numbers, by abelian subvarieties of $A_0$ of codimension at least $k$ under all isogenies between $A_0$ 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 PilaZannier strategy.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 DOI:
 10.48550/arXiv.1801.05701
 arXiv:
 arXiv:1801.05701
 Bibcode:
 2018arXiv180105701D
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 11G18;
 11G50;
 11U09;
 14G40;
 14K02
 EPrint:
 34 pages, accepted to JEMS, minor revisions