Seminar in Algebraic Geometry and Number Theory

Abstract: Let X be a curve over Q. We introduce a geometric non-degenerate criterion for X using certain projective models of X over Z called ico models. If X is non-degenerate, then X has genus >1 and we give an explicit Weil height bound for the Q-rational points of X. This establishes the effective Mordell conjecture for large classes of  (explicit) curves over Q. Moreover, on combining our height bound with Diophantine approximations, we can solve the ico analogue of the generalized Fermat problem. We will 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.