Unlikely intersections in families of abelian varieties and generalized Pell equations in polynomials 

Let A be an abelian scheme over a smooth irreducible curve S and let C be an irreducible curve inside A, where everything is defined over a number field k. We prove that if C is not contained in a proper subgroup scheme of A, then it intersects at most finitely many irreducible components of flat subgroup schemes of codimension at least 2. This is a special case of a more general conjecture of Pink and fits in the so called problems of unlikely intersections. This result has also some applications in the study of the almost-Pell equations in polynomials, generalising some previous results due to Masser and Zannier. This is a joint work with Fabrizio Barroero (Basel).