Collinear CM-points

Abstract

André's celebrated Theorem of 1998 implies that each complex straight line (apart from obvious exceptions) contains at most finitely many points whose both coordinates are j-invariants of elliptic curves with complex multiplication. 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 best possible.

Minor changes suggested by the referee

References in corpus (1)

• Six unlikely intersection problems in search of effectivity

Cited by in corpus (3)

• Geometry of PCF parameters in spaces of quadratic polynomials
• Some uniform effective results on André--Oort for sums of powers in ${\mathbb{C}}^n$
• Multiplicative independence of modular functions