Collinear CM-points

TL;DR

This paper proves that only finitely many complex lines contain more than two points with both coordinates as j-invariants of elliptic curves with complex multiplication, extending André's theorem.

Contribution

It establishes the finiteness of lines with multiple CM-point pairs, refining the understanding of CM points on complex lines.

Findings

Finitely many lines contain more than two CM points

Any two CM points determine a line with at most two such points

The result is optimal, as any two points determine a line

Abstract

Andr\'e'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.