  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.
