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Special Points on the Product of Two Modular Curves

Published online by Cambridge University Press:  04 December 2007

BAS EDIXHOVEN
BAS EDIXHOVEN
Affiliation:
Irmar, Campus de Beaulieu, 35042 Rennes cedex, France, e-mail: [email protected]
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Abstract

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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 C$^2$, 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 [ges ] 1such that X is the image, under the usual map, of the modular curve Y$_0$(m). The proof uses some number theory and some topological arguments.

Keywords

Shimura varietiescomplex multiplicationsubvarietieselliptic curvesmodular curves.
Type
Research Article
Information
Compositio Mathematica , Volume 114 , Issue 3 , December 1998 , pp. 307 - 320
Copyright
© 1998 Kluwer Academic Publishers