CM-Points on Straight Lines

Bill Allombert (1, 2) , Yuri Bilu (1) , Amalia Pizarro-Madariaga (3)

1 IMB - Institut de Mathématiques de Bordeaux
2 LFANT - Lithe and fast algorithmic number theory
3 Departamento de Matematica [Valparaiso]

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.

Domaines: Théorie des nombres [math.NT]

Bill Allombert, Yuri Bilu, Amalia Pizarro-Madariaga. CM-Points on Straight Lines. Analytic Number Theory : In Honor of Helmut Maier’s 60th Birthday, 2015, 978-3-319-22239-4 ⟨10.1007/978-3-319-22240-0_1⟩. ⟨hal-01114121⟩