Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Wednesday, 1 December 2004, 13:15 - 15:30 (with a 15 min break), room 405

Pierre Parent and Andrei Yafaev: Rational points on Shimura curves

Abstract:

We will present some methods to study the rational points of certain modular curves, and more general Shimura curves. In both situations the moduli interpretation of these spaces shows that they are endowed with "trivial" rational points, coming from complex multiplication. In certain cases our methods allow to prove that the whole set of rational points of the curves is made of trivial points (or cusps). These methods rely in particular on the precise description of special fibres of our curves, and an arithmetic interpretation of well-known formulae for special values of L-functions.