Lecture 1: The games of Steiner and Poncelet and algebraic group schemes

Umberto Zannier, Scuola Normale Superiore-Pisa
McDonnell Hall A02

We shall briefly present in very elementary terms the 'games' of Steiner and Poncelet, amusing mathematical solitaires of the XIX Century, also related to elliptic billiards.  We shall recall that the finiteness of the game is related to torsion in tori or elliptic curves. We shall illustrate how one can vary the data of the games, obtaining families of elliptic curves and sections on elliptic schemes, for which we seek torsion values. This is related to the so-called `Betti-map', which we shall describe.

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