Publication details

Equations in three singular moduli

authored by: Guy Fowler
Abstract: Let a∈Z_>0 and ϵ₁,ϵ₂,ϵ₃∈{±1}. We classify explicitly all singular moduli x₁,x₂,x₃ satisfying either ϵ₁x₁^a+ϵ₂x₂^a+ϵ₃x₃^a∈Q or (x₁^ϵ₁x₂^ϵ₂x₃^ϵ₃)^a∈Q^×. In particular, we show that all the solutions in singular moduli x₁,x₂,x₃ to the Fermat equations x₁^a+x₂^a+x₃^a=0 and x₁^a+x₂^a−x₃^a=0 satisfy x₁x₂x₃=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.
Organisation(s): Institute of Algebra, Number Theory and Discrete Mathematics
Type: Article
Journal: Journal of number theory
Volume: 243
Pages: 256-297
No. of pages: 42
ISSN: 0022-314X
Publication date: 02.2023
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Algebra and Number Theory
Electronic version(s): https://doi.org/10.48550/arXiv.2105.12696 (Access: Open)
https://doi.org/10.1016/j.jnt.2022.09.012 (Access: Closed)