Special Points on the Product of Two Modular Curves |
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Authors: | Bas Edixhoven |
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Affiliation: | (1) Irmar, Campus de Beaulieu, 35042 Rennes cedex, France |
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Abstract: | We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C2, containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m 1such that X is the image, under the usual map, of the modular curve Y20(m). The proof uses some number theory and some topological arguments. |
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Keywords: | Shimura varieties complex multiplication subvarieties elliptic curves modular curves. |
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