## On the André-Oort conjecture for Hilbert modular surfaces

Prépublication de l'I.R.M.A.R. 99-59

### Summary

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.

fichier DVI/DVI file (.dvi)
(180kB)

fichier PostScript comprimé/compressed PostScript file (.ps.gz)
(147kB)

fichier PostScript non comprimé/uncompressed PostScript file (.ps)
(392kB)

fichier PDF non comprimé/uncompressed PDF file (.pdf)
(299kB)

Retour à mes publications.

###### Dernière modification: 12/11/1999

edix@maths.univ-rennes1.fr