We shall consider elliptic pencils, of which the best-known example is probably the Legendre family Lt: y2=x(x-1)(x–t) where t is a parameter. Given a section P(t) (i.e. a family of points on Lt depending on t) it is an issue to study the set of complex b such that P(b) is torsion on Lb. We shall recall a number of results on the nature of this set. Then we shall present some applications (obtained jointly with P. Corvaja) to elliptical billiards. For instance, if two players hit the same ball with directions forming a given angle in (0,𝞹), there are only finitely many cases for which both billiard trajectories are periodic.
This video is part of the Number Theory Web Seminar series.
