# Unlikely intersections with isogeny orbits

Dill, Gabriel Andreas. Unlikely intersections with isogeny orbits. 2019, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_13523

Chapter 3 contains a characterization of curves in abelian schemes, defined over $\bar{\mathbb{Q}}$, that intersect certain (enlarged) isogeny orbits infinitely often. An isogeny orbit is the set of all images of a fixed finite-rank subgroup of a fixed abelian variety, both of which we assume to be defined over $\bar{\mathbb{Q}}$, under all isogenies between the fixed abelian variety and some fiber of the abelian scheme. It is enlarged (depending on a parameter $k$) if the finite-rank subgroup is replaced by the union of its translates by abelian subvarieties of codimension at least $k$. The obtained characterization yields a stronger version of the so-called André-Pink-Zannier conjecture for curves in the case where everything is defined over $\bar{\mathbb{Q}}$.
Chapter 4 contains a characterization of subvarieties of arbitrary dimension of abelian schemes, defined over $\bar{\mathbb{Q}}$, that intersect a (non-enlarged) isogeny orbit in a Zariski dense set, under technical restrictions on the abelian scheme and the fixed abelian variety. The restrictions are satisfied for example if the abelian scheme is a fibered power of a non-isotrivial elliptic scheme and the fixed abelian variety is a power of an elliptic curve without CM that is defined over $\bar{\mathbb{Q}}$. This again proves a stronger version of the André-Pink-Zannier conjecture in certain cases. The proof combines the Pila-Zannier method with a generalized Vojta-Rémond inequality.
Chapter 5 contains (among other results) a characterization of semiabelian schemes over a curve, defined over $\bar{\mathbb{Q}}$, with infinitely many pairwise isogenous fibers. It also contains an extension of the approach to the Manin-Mumford conjecture through use of the Galois action (developed and applied by Serre, Tate, Lang, and Hindry) to the problem of studying torsion points on pairwise isogenous fibers in abelian schemes.
Chapter 6 consists of joint work with Fabrizio Barroero, where we show that the Zilber-Pink conjecture for complex abelian varieties follows from the same conjecture for abelian varieties defined over $\bar{\mathbb{Q}}$. Moreover, the conjecture holds for a curve in a complex abelian variety and it holds in any complex abelian variety that contains no abelian subvariety of dimension larger than $4$ that can be defined over $\bar{\mathbb{Q}}$.