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An effective “Theorem of André” for CM-points on a plane curve

Published online by Cambridge University Press:  04 October 2012

YURI BILU ,
DAVID MASSER  and
UMBERTO ZANNIER
YURI BILU
Affiliation:
Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France. e-mail: yuri@math.u-bordeaux1.fr
DAVID MASSER
Affiliation:
Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland. e-mail: David.Masser@unibas.ch
UMBERTO ZANNIER
Affiliation:
Scuola Normale Superiore, Piazza Cavalieri 7, 56126 Pisa, Italy. e-mail: u.zannier@sns.it
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Abstract

It is a well known result of Y. André (a basic special case of the André-Oort conjecture) that an irreducible algebraic plane curve containing infinitely many points whose coordinates are CM-invariants is either a horizontal or vertical line, or a modular curve Y0(n). André's proof was partially ineffective, due to the use of (Siegel's) class-number estimates. Here we observe that his arguments may be modified to yield an effective proof. For example, with the diagonal line X1+X2=1 or the hyperbola X1X2=1 it may be shown quite quickly that there are no imaginary quadratic τ1,τ2 with j(τ1)+j(τ2)=1 or j(τ1)j(τ2)=1, where j is the classical modular function.

Keywords

11G3011G1511G18
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 1 , January 2013 , pp. 145 - 152
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

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