Equations with powers of singular moduli

Abstract

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(τ ), j(τ ′ ) such that the numbers 1, j(τ ) m and j(τ ′ ) n are linearly dependent over Q for some positive integers m, n, must be of degree at most 2. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in C 2 defined over Q. On the other hand, we show that, with "obvious" exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to an hyperbola xy = A, where A ∈ Q.

Fields generated by sums and products of singular moduli

“…In particular, [Q(x) : Q] = h(∆ x ). For all details, see, for instance, [6, §7 and §11] Starting from the ground-breaking article of André [2] equations involving singular moduli were studied by many authors, see [1,4,8] for a historical account and further references. In particular, Kühne [7] proved that equation x + y = 1 has no solutions in singular moduli x and y, and Bilu et al [5] proved the same for the equation xy = 1.…”

Linear Independence of Powers of Singular moduli of Degree Three

“…Let j(τ) be a singular modulus of discriminant ∆. It is well known that the conjugates of j(τ) over Q can be described explicitly (see, for instance, [10,Subsection 2.2]). In particular, j(τ) admits one real conjugate which has the property that it is much larger in absolute value than all its other conjugates, called the dominant j-value of discriminant ∆.…”

“…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.…”

“…In [10], Riffaut generalised Theorem 1.1 by introducing exponents; that is, instead of Ax + By = C, he considered the more general equation Ax m + By n = C, where the positive integer exponents m, n are unknown as well. He proved that, if x y, then x, y generate the same number field of degree h ≤ 3 and h = 3 is possible only if either…”

Trinomials with given roots

“…This problem emerged in the work of Riffaut [18] on effective André-Oort. We invite the reader to consult the article of Riffaut for more context and motivation.…”