Non-degenerate Diophantine equations

Abstract: We present explicit height bounds for the solutions of Diophantine equations satisfying a certain non-degeneracy criterion. In particular our results establish the effective Mordell conjecture for large classes of (explicit) curves over the rational numbers. We illustrate the non-degeneracy criterion for various classical equations and we explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser-Wustholz isogeny estimates. Joint work with Shijie Fan.