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Unlikely Intersections on families of abelian varieties (Part I) Barroero, Fabrizio


Let $A$ 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 $k$. Then $C$ defines $n$ points $P_1, ... P_n$ points on $A(k(C))$. Then, there are at most finitely many points $c$ on the curve such that the specialized $P_1(c), ... P_n(c)$ 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. The proof of these theorems uses a method introduced by Pila and Zannier and combines results coming from o-minimality with some Diophantine ingredients. These results have applications to the study of the solvability of some Diophantine equations in polynomials.

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