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Rational points of modular curves: an arakelovian point of view

Banff International Research Station for Mathematical Innovation and Discovery

General methods from diophantine geometry have been very successful in proving finiteness results for points of algebraic curves with values in number fields. Those results however are generally not effective, for deep reasons, and this prevents from proving triviality (and not only finiteness) of relevant sets of rational points. In this talk I will explain how the situation can be much better in the case of modular curves because of the only, deep, but simple feature that in many cases their jacobian possesses a non-trivial quotient with rank zero over the rationals. From that we can derive effective upper bounds for the height of rational points by using specific arakelovian tools.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: Universite de Bordeaux

2017-11-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63721

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/