Modularity and integral points on moduli schemes

Abstract: In this talk we present new Diophantine applications of modularity results. In the first part, we use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. On working out the method for moduli schemes corresponding to Mordell equations, we improve the actual best explicit height bounds for Mordell equations. In the second part, we combine Faltings’ method with Serre’s modularity conjecture to establish the effective Shafarevich conjecture for abelian varieties of (product) GL2-type and then we discuss applications to the effective study of integral points on certain higher dimensional moduli schemes (e.g. Hilbert modular varieties).