Let $A\rightarrow S$ be a non-isotrivial family of abelian varieties over a smooth irreducible curve $S$. Suppose the generic fiber of $A$ is simple and call $R$ its endomorphism ring. We consider an irreducible curve $C$ in the $n$-fold fibered power of $A$ and suppose that everything is defined over a number field $k$. This defines $n$ points $P_1, \dots , P_n$ in $A(k(C))$. Then, there are at most finitely many $c \in C(\mathbb{C})$ such that the specialised $P_1(c), \dots, P_n(c)$ are dependent over $R$, unless they were already identically dependent. This, combined with earlier works of the authors and of Habegger and Pila, gives a general unlikely intersections statement for (not necessarily simple) families of abelian varieties. We will conclude by talking about applications of this statement to some polynomial Diophantine equations.

This is joint work with L. Capuano