On the Andre-Oort conjecture for Hilbert modular surfaces

Details On the Andre-Oort conjecture for Hilbert modular surfaces On the Andre-Oort conjecture for Hilbert modular surfaces

1999-11-15

arxiv.org/abs/math/9911272v1

Mathematics

Number Theory

Scientific paper

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.

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