### On the Andre-Oort conjecture for Hilbert modular surfaces, by Bas Edixhoven

We prove, assuming the generalized Riemann hypothesis, the Andre-Oort
conjecture for Hilbert modular surfaces. More precisely, let K be a real
quadratic field and let S be the coarse moduli space of complex abelian
surfaces with multiplications by the ring of integers of K. Let C be an
irreducible closed curve in S, and suppose that C contains infinitely many
complex multiplication points. Then we prove, assuming GRH, that C is of
Hodge type, meaning, in this case, that it parametrizes abelian varieties
with more endomorphisms. Also, if we assume that C has infinitely many CM
points that correspond to abelian surfaces that lie in one isogeny class,
we prove that C is of Hodge type without assuming GRH. This last result is
motivated by applications by Wolfart, Cohen and Wustholz.

Bas Edixhoven <edix@maths.univ-rennes1.fr>