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Unlikely Intersections in certain families of abelian varieties and the polynomial Pell equation

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Given n independent points on the Legendre family of elliptic curves of equation Y^2=X(X-1)(X-t) with coordinates algebraic over Q(t), we will see that there are at most finitely many specializations of c such that two independent relations hold between the n points on the specialized curve. This fits in the framework of the so-called Unlikely Intersections. We will then see an higher-dimensional analogue of this result and explain how it applies to the problem of studying the solvability of the (almost-)Pell equation in polynomials. This is joint work with Laura Capuano.

This talk is part of the Number Theory Seminar series.

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