Torsion in elliptic familes and applications to billiards

Abstract: 

We shall consider elliptic pencils, of which the best-known example is probably the Legendre family LtLt: y2=x(x−1)(x−t)y2=x(x−1)(x−t) where tt is a parameter. Given a section P(t)P(t) (i.e. a family of  points on LtLt depending on tt)  it is an issue to study the set of complex bb such that P(b)P(b) is torsion on LbLb. 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)(0,π), there are only finitely many cases for which both billiard trajectories are periodic.

This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.

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Organisers:

Mike Bennett (University of British Columbia)

Philipp Habegger (University of Basel)

Alina Ostafe (UNSW Sydney)