Modularity and integral points on moduli schemes

Abstract: On combining modularity with Arakelov theory, we obtain finiteness results for integral points on moduli schemes of elliptic curves. The method is fully effective. For example, on applying the method to the projective line minus three points and to once punctured Mordell elliptic curves, we improve the actual best explicit height upper bounds for the solutions of S-unit and Mordell equations. In addition, the method gives an effective Shafarevich conjecture for abelian varieties of GL(2)-type.