

Globus Seminar
April 25, 2013 15:40, Moscow, IUM (Bolshoi Vlas'evskii per., 11)






On the size of generators of solutions of some Diophantine equations
M. Hindry^{ab} ^{a} Université Paris VII – Denis Diderot
^{b} Laboratoire J.V. Poncelet, Independent University of Moscow

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Abstract:
It has been known since at least Fermat that the set of integral solutions to the equation x^2dy^2=1 form a finitely generated group of rank one. It has been known since at least Poincaré that the set of rational solutions to equations of the type y^2=x^3+ax+b form a group; in fact, as Mordell proved, the latter group is also finitely generated.
There is a natural notion of size or height of solutions, so an important and natural question is to estimate the minimal size of a set of generators. The questions can easily be generalized on one hand to the group of units of a number field and, on the other hand, to the group of rational points of an abelian variety over a global field.
The answer for the first case is essentially known, though there are important unsettled related questions; the answer for the second case is essentially conjectural. We will discuss what we know, conjecture and give examples where theorems may be proven. This will take us to a journey through some arithmetic geometry, zeta functions etc., i.e. several number theorists favourite toys.

