Integral points on moduli schemes

Abstract: The strategy of combining the method of Faltings (Arakelov, Paršin, Szpiro) with modularity and Masser–Wüstholz isogeny estimates allows to explicitly bound the height and the number of the solutions of certain Diophantine equations related to integral points on moduli schemes of abelian varieties. In this paper we survey the development and various applications of this strategy.