No singular modulus is a unit

TL;DR

This paper provides an effective proof that no singular modulus is an algebraic unit, improving upon previous non-effective results by explicitly bounding discriminants and ruling out all cases through computational methods.

Contribution

It offers the first effective proof that singular moduli cannot be algebraic units, with explicit bounds and computational verification.

Findings

No singular modulus is an algebraic unit.

Explicit discriminant bound less than 10^{15}.

Computational methods exclude all remaining cases.

Abstract

A result of the second-named author states that there are only finitely many CM-elliptic curves over $\mathbb{C}$ 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 $10^{15}$. 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 $\mathbb{C}^n$ not containing any special points.