On the André-Oort conjecture for Hilbert modular surfaces

Prépublication de l'I.R.M.A.R. 99-59


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.

This article will appear in the proceedings of the 1999 Texel conference on arithmetic and abelian varieties. The version here has an appendix of 19 pages containing what can be seen as the author's scratch paper. It is included here with the idea that it might make reading of the article somewhat easier.

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Dernière modification: 12/11/1999