On the quantitative Manin-Mumford conjecture

The Manin-Mumford conjecture, proved by Raynaud in 1983, states that there are only finitely many torsion points on the embedding of a curve of genus g>1 into its Jacobian. In the first part of the talk, I will show that over function fields there are at most 16g²+32g+124 such points. This is joint work with Looper and Silverman. In the second part, I will discuss recent progress on an analogous result in the case of number fields.