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

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 (j(τ),j(τ)), where τ,τ 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.

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CM points, André–Oort
Mathematical Subject Classification 2010
Primary: 11G15
Secondary: 11G18
Received: 2 January 2016
Revised: 27 November 2016
Accepted: 31 March 2017
Published: 12 July 2017
Yuri Bilu
Institut de Mathématiques de Bordeaux
Université de Bordeaux et CNRS
Florian Luca
School of Mathematics
University of the Witwatersrand
South Africa Max Planck Institute for Mathematics
David Masser
Mathematisches Institut
Universität Basel