# No singular modulus is a unit

Bilu, Yuri and Habegger, Philipp and Kühne, Lars. (2018) No singular modulus is a unit. International Mathematics Research Notices. rny274.

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Official URL: https://edoc.unibas.ch/68925/

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.