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Marc Hindry

Diophantine Geometry (Диофантова геометрия)

The purpose of the course is to introduce basic tools from algebraic geometry (abelian varieties, line bundles, etc) and algebraic number theory (heights) used in Diophantine Geometry and apply to describe classical theorems of Mordell-Weil : "the group of rational points of an abelian variety is finitely generated" and Faltings : "the set ot rational point of a subvariety of an abelian variety is not Zariski dense (outside trivial cases)"

Program:

1. Geometry of curves and algebraic geometry

2. Heights (Weil, Nron-Tate, Arakelov)

3. Mordell-Weil theorem

4. Faltings theorems (sketch)

References:

• Lang : Fundamentals of Diophantine Geometry, Springer 1983.
• Serre : Lectures on the Mordell-Weil theorem, Vieweg 1989.
• Hindry-Silverman : Diophantine Geometry: an Introduction, Springer 2000.
• Bombieri-Gubler : Heights in Diophantine Geometry, Cambridge University Press 2006.