Umberto Zannier (Scuola Normale Superiore, Italy)

Abstract: After a very brief review of basics on elliptic curves and their families, we shall consider “sections” of such  families, and especially their “torsion values”.  For instance, what can be said of the complex numbers b for which [latex](2, \sqrt{2(2-b)})[/latex] is torsion on the Legendre curve [latex]y^2=x(x-1)(x-b)[/latex]? In particular, we shall recall results of “Manin-Mumford type” and focus to illustrate some applications to elliptical billiards. Finally, if time allows we shall frame these issues as special cases of a general question in arithmetic dynamics, which can be treated with different methods, depending on the context. (Most results refer to work with Pietro Corvaja and David Masser.)