Seminari del Dipartimento
Geometria The uniform Bogomolov conjecture for algebraic curves Lars Kühne 15-09-2022 - 14:15 Largo San Leonardo Murialdo,1 - Pal.C
I will present an equidistribution result for families of (non-degenerate) subvarieties in a family of abelian varieties. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Furthermore, one can deduce a rather uniform version of the Mordell conjecture by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick–Brown). All these results have been recently generalized beyond curves in joint work with Ziyang Gao and Tangli Ge, but I will focus on the simpler case of curves. |