Computing integral points on the moduli space of smooth quartic curves
In 1983 Faltings proved that the set of S-integral points on the moduli space of smooth curves of any given genus > 1 is finite. As an application, he deduced that the Mordell conjecture: the set of rational points on a curve of genus > 1 is finite. Although some parts of Faltings's proof are effective, the overall proof remains ineffective. In this talk, I will explain how to effectively compute the integral points on the moduli space of genus three smooth quartic curves by using the cyclic covering trick, the geometry of Del Pezzo surfaces, and the well-known fact that we can solve S-unit equations in number fields effectively (due to Baker-Wustholz, Evertse-Gyory).