Algebra & Number Theory

Collinear CM-points

Yuri Bilu, Florian Luca, and David Masser

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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 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

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

Received: 2 January 2016
Revised: 27 November 2016
Accepted: 31 March 2017
First available in Project Euclid: 12 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]

CM points André–Oort


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.

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