Special points on products of modular curves
arXiv:math/0302138 · doi:10.1215/S0012-7094-04-12624-7
Abstract
We prove the Andre-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let n be a positive integer, and let S be a subset of C^n (with C the complex numbers) consisting of points all of whose coordinates are j-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreducible components of the Zariski closure of S are ``special subvarieties'', i.e., determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant is proved unconditionally.
21 pages, referee's remarks have been taken into account, some references updated, to appear in Duke Mathematical Journal
References in corpus (1)
Cited by in corpus (7)
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- Galois orbits and equidistribution: towards the Andre-Oort conjecture
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- Linear Equations in Singular Moduli
- The Andre-Oort conjecture
- Geometry of PCF parameters in spaces of quadratic polynomials
- Product Formulas for Periods of CM Abelian Varieties and the Function Field Analog