1999/11/15 by Bas Edixhoven, Edixhoven, Bas · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.math/9911272
arxiv created 1999/11/15 · openalex publication_date 1999/11/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.