Recent progress in diophantine geometry

Abstract

Diophantine problems have a long history. I review recent results, due to P. Vojta, G. Frey and myself. In the first half of this century, Thue and Siegel developed the method of Diophantine approximation to prove finiteness results for the number of rational or integral points on certain curves. However, they did not manage to prove the Mordell conjecture, which in some sense is the strongest possible statement of this kind. This has been achieved only recently by P. Vojta, and his methods allow generalisations which settle the question for arbitrary subvarieties of Abelian varieties. However, before that, Arakelov and Parshin had developed other methods which work for function fields, and I myself had managed to prove the conjecture with them. Also recently Masser and Wüstholz have used methods from transcendence theory to give effective versions of some results.

I intend to review these and also explain on the way some nice mathematics which has come out of this, namely Arakelov theory. Finally, I intend to cover the relation between Fermat's conjecture and elliptic curves, which was discovered by G. Frey.

Text by the author, supplemented by Enric Nart with material from the videotape of the talk.

Fields Medal 1986 for his proof of the Mordell, Shafarevich and Tate conjectures using methods of arithmetic algebraic geometry to compute the height of the Abelian varieties considered as points in a suitable space.