Integral points on moduli schemes of elliptic curves

Abstract

We combine the method of Faltings (Arakelov, Paršin, Szpiro) with the Shimura–Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example S ‐unit and Mordell equations, this gives an effective method which does not rely on Diophantine approximation or transcendence techniques.