2018/05/18 by Yu. Bilu, Bilu, Yu., Philipp Habegger +3 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.1805.07167
openalex publication_date 2018/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A result of the second-named author states that there are only finitely many CM-elliptic curves over ℂ whose j-invariant is an algebraic unit. His proof depends on Duke's Equidistribution Theorem and is hence non-effective. In this article, we give a completely effective proof of this result. To be precise, we show that every singular modulus that is an algebraic unit is associated with a CM-elliptic curve whose endomorphism ring has discriminant less than 1015. Through further refinements and computer-assisted computations, we eventually rule out all remaining cases, showing that no singular modulus is an algebraic unit. This allows us to exhibit classes of subvarieties in ℂn not containing any special points.