Special points on products of modular curves

Details Special points on products of modular curves Special points on products of modular curves

2003-02-12

arxiv.org/abs/math/0302138v2

Duke Math. J. 126 (2005), no. 2, 325--348.

Mathematics

Number Theory

21 pages, referee's remarks have been taken into account, some references updated, to appear in Duke Mathematical Journal

Scientific paper

10.1215/S0012-7094-04-12624-7

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.

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