Collinear CM-points

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.