#### Vol. 11, No. 5, 2017

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Collinear CM-points

### Yuri Bilu, Florian Luca and David Masser

Vol. 11 (2017), No. 5, 1047–1087
##### Abstract

André’s celebrated theorem of 1998 implies that each complex straight line $Ax+By+C=0$ (apart from obvious exceptions) contains at most finitely many points $\left(j\left(\tau \right),j\left({\tau }^{\prime }\right)\right)$, where $\tau ,{\tau }^{\prime }\in ℍ$ are algebraic of degree $2$. We show that there are only a finite number of such lines which contain more than two such points. As there is a line through any two complex points, this is the best possible result.

##### Keywords
CM points, André–Oort
Primary: 11G15
Secondary: 11G18
##### Milestones
Received: 2 January 2016
Revised: 27 November 2016
Accepted: 31 March 2017
Published: 12 July 2017
##### Authors
 Yuri Bilu Institut de Mathématiques de Bordeaux Université de Bordeaux et CNRS Talence France Florian Luca School of Mathematics University of the Witwatersrand Johannesburg South Africa Max Planck Institute for Mathematics Bonn Germany David Masser Mathematisches Institut Universität Basel Basel Switzerland