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