Umberto Zannier: Torsion in elliptic familes and applications to billiards (NTWS 030)
Channel: Number Theory Web Seminar
Date: 2023-06-04

Abstract: We shall consider elliptic pencils, of which the best-known example is probably the Legendre family L_t: y^2=x(x-1)(x-t) where t is a parameter. Given a section P(t) (i.e. a family of  points on L_t depending on t)  it is an issue to study the set of complex b such that P(b) is torsion on L_b. 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,\pi), there are only finitely many cases for which both billiard trajectories are periodic.

Link to slides: https://drive.google.com/file/d/1DBTjhStHFzewzzfXfDLPC0Vp43VdRXtB

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Tuesday, September 1, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm NZST)