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Transcendental methods in number theory and diophantine problems

Final Activity Report Summary - TRANSCDIOPH (Transcendental methods in number theory and diophantine problems)

My research concerns Diophantine geometry, this branch of Mathematics deals with integral or rational solutions to polynomial equations. More precisely, my research evolved around three fundamental finiteness theorems: the theorems of Mordell-Weil (the group of rational points of an abelian variety is finitely generated), Siegel's theorem (the integral points of a non-exceptional algebraic curve are finite) and Faltings' theorem (the rational points of a curve of genus 2 are finite). An important aspect of these theorems and which remains an open problem, is the problem of the effectivity: how can one find the points?

An argument which often returns in Diophantine geometry and which, in particular appears or can appear as part of the proof of the theorems mentioned above, is the following. Given a special map between two algebraic objects, translate information to one object from information in the other, easier to obtain. The Chevalley-Weil theorem gives an answer in the case of the étale coverings of algebraic varieties. The goal was to obtain, in collaboration with Yuri Bilu (Bordeaux) and Marco Strambi (Pisa) a 'quantitative' version of the Chevalley-Weil theorem for curves. We give a completely effective result, both in the projective and the affine case.

I was also interested on the 'size' of the Mordell-Weil and Tate-Shafarevich groups of an abelian variety. I obtained conjectural bounds for the order of the Tate-Shafarevich group and for the canonical height for the elements of a basis of the Mordell-Weil group. The results rest on the Birch and Swinnerton-Dyer conjecture (BSD-conjecture for short). The BSD-conjecture is one of the seven problems of the Clay Mathematics Institute and predicts a better understanding of abelian varieties; it translate analytic information into geometric or arithmetic one. My results improve some previous results. They rest on conjectures certainly strong, but from now on classical.

The remainder of the project connects different branches of number theory: Baker's method, the abc-conjecture and the BSD-conjecture. Alan Baker was awarded with the Fields Medal for his result on linear forms in logarithms. The abc-conjecture of Masser-Oesterlé implies strong / effective versions of important theorems, as, for example, Siegel's and Fatings' theorems. In a joint work with Vincent Bosser (Caen), using the elliptic analogue of Baker's method, we proved that the BSD-conjecture for a single elliptic curve would imply a result in direction of the abc-conjecture over number fields.