Papers › Triples of singular moduli with rational product
Triples of singular moduli with rational product
Guy Fowler
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
We show that all triples (x₁,x₂,x₃) of singular moduli satisfying x₁ x₂ x₃ ∈ℚ^× are "trivial". That is, either x₁, x₂, x₃ ∈ℚ; some xᵢ ∈ℚ and the remaining xⱼ, xₖ are distinct, of degree 2, and conjugate over ℚ; or x₁, x₂, x₃ are pairwise distinct, of degree 3, and conjugate over ℚ. This theorem is best possible and is the natural three dimensional analogue of a result of Bilu, Luca, and Pizarro-Madariaga in two dimensions. It establishes an explicit version of the Andr\'e--Oort conjecture for the family of subvarieties V_α ⊂ℂ³ defined by an equation x₁ x₂ x₃ = α∈ℚ.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections