Some problems of unlikely intersections in families of abelian varieties

This thesis investigates several instances of the Zilber–Pink conjecture, originally formulated by Zilber and independently by Bombieri, Masser and Zannier for tori, and later generalized by Pink in the framework of mixed Shimura varieties. The conjecture, which unifies and extends fundamental results in arithmetic geometry, has been intensively studied over the last two decades. Our focus lies on the case of curves in families of abelian varieties. First, for curves in products of fibered powers of elliptic schemes, we prove that such a curve contains only finitely many points lying in an algebraic subgroup of a fiber for which there exist non-trivial homomorphisms between the two powers, extending previous results of Masser-Zannier and Barroero-Capuano. Second, for curves in non-isotrivial abelian schemes $A \to S$, we prove that if $C \subset A$ is not contained in a fiber nor in a translate of a flat subgroup scheme, then $C$ meets the union of all proper algebraic subgroups of the CM fibers of $A$ in at most finitely many points. This generalizes a previous result by Barroero for fibered powers of elliptic schemes to higher-dimensional abelian varieties. Both results are obtained using the Pila–Zannier strategy, combining functional transcendence and o-minimality with new arithmetic estimates for canonical heights of images of points under endomorphisms in abelian varieties.