CM-Points on Straight Lines

Abstract

International audienceWe 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

Linear Equations in Singular Moduli

“…Up to now, all non-trivial effectively solvable cases ( [1,2,3,11,12]) of the André-Oort conjecture have been restricted to the case of curves. In particular, the only known examples of algebraic subvarieties in Shimura varieties that are known to contain no special points are either curves or weakly special subvarieties.…”

Decompositions of singular abelian surfaces

“…In most cases the field L is precisely the composite of L 1 and L 2 , as it is stated in the following Proposition 2.5 (Proposition 3.1 in [1]). Assume all conditions above are satisfied.…”

Linear Independence of Powers of Singular moduli of Degree Three

“…In particular, Kühne [8] proved that the equation x + y = 1 has no solutions in singular moduli x, y and Bilu et al [5] proved that the same conclusion holds for the equation xy = 1. These results were generalised in [1] and [4]. In [1], solutions of all linear equations Ax + By = C, with A, B, C ∈ Q, were determined.…”

“…These results were generalised in [1] and [4]. In [1], solutions of all linear equations Ax + By = C, with A, B, C ∈ Q, were determined. The main result of [1] is the following theorem.…”

“…Starting from the ground-breaking article of André [2], equations involving singular moduli were studied by many authors (see [1,4,10] for a historical account and further references). In particular, Kühne [8] proved that the equation x + y = 1 has no solutions in singular moduli x, y and Bilu et al [5] proved that the same conclusion holds for the equation xy = 1.…”