CM-Points on Straight Lines
Bill Allombert, Yuri Bilu, Amalia Pizarro-Madariaga
TL;DR
This paper proves that, generally, CM-points (special algebraic points related to elliptic curves) do not lie on straight lines over Q in the complex plane, extending previous specific results to a broader context.
Contribution
The paper generalizes Kuhne's result by showing that CM-points rarely lie on straight lines over Q, except for obvious special cases.
Findings
CM-points generally do not lie on straight lines over Q
The result extends Kuhne's specific line case to broader lines
Identifies conditions under which CM-points can lie on lines
Abstract
We prove that, with "obvious" exceptions, a CM-point (j(\tau),j(\tau')) cannot belong to a straight line in C^2 defined over Q. This generalizes a result of K\"uhne, who proved this for the line x+y=1.
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Taxonomy
TopicsMathematics and Applications · Geometric Analysis and Curvature Flows · History and Theory of Mathematics