Data Entry: Please note that the research database will be replaced by UNIverse by the end of October 2023. Please enter your data into the system https://universe-intern.unibas.ch. Thanks
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 arguments, 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.
