Papers › Equations in three singular moduli: the equal exponent case
Equations in three singular moduli: the equal exponent case
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.
Let a ∈ℤ_(>0) and ϵ₁, ϵ₂, ϵ₃ ∈{±1}. We classify explicitly all singular moduli x₁, x₂, x₃ satisfying either ϵ₁ x₁ᵃ + ϵ₂ x₂ᵃ + ϵ₃ x₃ᵃ ∈ℚ or (x₁^(ϵ₁) x₂^(ϵ₂) x₃^(ϵ₃))ᵃ ∈ℚ^×. In particular, we show that all the solutions in singular moduli x₁, x₂, x₃ to the Fermat equations x₁ᵃ + x₂ᵃ + x₃ᵃ= 0 and x₁ᵃ + x₂ᵃ - x₃ᵃ= 0 satisfy x₁ x₂ x₃ = 0. Our proofs use a generalisation of a result of Faye and Riffaut on the fields generated by sums and products of two singular moduli, which we also establish.
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