Equations with powers of singular moduli

Antonin Riffaut (1, 2)

1 IMB - Institut de Mathématiques de Bordeaux
2 LFANT - Lithe and fast algorithmic number theory

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.

Keywords: Singular modulus; Conjecture of André–Oort; Complex multiplication

Domains: Number Theory [math.NT]

Antonin Riffaut. Equations with powers of singular moduli. International Journal of Number Theory, 2019, 15 (3), pp.445-468. ⟨10.1142/S1793042119500234⟩. ⟨hal-01630363⟩