**Title:** Dynamics and small points in families of elliptic curves

**Speaker:** Niki Myrto Mavraki

**Speaker Info:** University of British Columbia

**Brief Description:**

**Special Note**:

**Abstract:**

In a paper by Lang in 1965, it was proved that if an irreducible curve given by the zero locus of a polynomial f(x,y) contains infinitely many points with both coordinates roots of unity, then f(x,y)=x^ny^m-a, for integers n and m and a root of unity a. Subsequently, in 1992 Zhang proved the so-called Bogomolov extension of Lang's theorem. It asserts that the conclusion of the theorem remains true if the curve contains infinitely many points of `small height'. These results were generalized to abelian varieties. More recently, Masser and Zannier proved a result in the spirit of Lang's theorem, concerning torsion points in a family of products of elliptic curves.We place these results in the context of dynamics and prove a Bogomolov-type extension of the Masser-Zannier result. To do so we use the work of Masser and Zannier, combined with Silverman's work on the variation of canonical heights on an elliptic family and an equidistribution theorem by Chambert-Loir, Thuillier and Yuan. This is joint work with Laura DeMarco.

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