## Algebra & Number Theory

### 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 $(j(τ),j(τ′))$, where $τ,τ′∈ ℍ$ 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.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 5 (2017), 1047-1087.

Dates
Revised: 27 November 2016
Accepted: 31 March 2017
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.ant/1513090722

Digital Object Identifier
doi:10.2140/ant.2017.11.1047

Mathematical Reviews number (MathSciNet)
MR3671431

Zentralblatt MATH identifier
06748166

Keywords
CM points André–Oort

#### Citation

Bilu, Yuri; Luca, Florian; Masser, David. Collinear CM-points. Algebra Number Theory 11 (2017), no. 5, 1047--1087. doi:10.2140/ant.2017.11.1047. https://projecteuclid.org/euclid.ant/1513090722

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