No Singular Modulus Is a Unit

A result of the 2nd-named author states that there are only finitely many complex multiplication (CM)-elliptic curves over C whose j-invariant is an algebraic unit. His proof depends on Duke's equidistribution theorem and is hence noneffective. 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 C-n not containing any special points.