Integrality properties in the moduli space of elliptic curves: CM case

Abstract

For each algebraic number [Formula: see text], a result of Habegger [P. Habegger, Singular moduli that are algebraic units, Algebra Number Theory 9(7) (2015) 1515–1524] shows that there are only finitely many singular moduli [Formula: see text] such that [Formula: see text] is an algebraic unit. His result uses Duke’s Equidistribution Theorem and is thus not effective. In this paper, we give an effective proof of Habegger’s result assuming that [Formula: see text] is not a singular modulus itself. We give an explicit bound, which depends only on [Formula: see text], on the discriminant [Formula: see text] associated with a singular modulus [Formula: see text] such that [Formula: see text] is a unit. This implies explicit bounds on the number of these singular moduli.