  Following N. Elkies ("ABC implies Mordell") we show that the abc conjecture
of Masser-Oesterle implies an effective version of Siegel's theorem about
integral points on algebraic curves, i.e. an upper bound for the S-integral
points where the dependence on S is explicit. The converse statement is also
announced in this note. For both results, the main geometric tool is a theorem
of G.V. Belyi.

Number Theory