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

摘要

For each algebraic number alpha is an element of(Q) over bar, 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 j such that j - alpha 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 alpha is not a singular modulus itself. We give an explicit bound, which depends only on alpha, on the discriminant Delta associated with a singular modulus j such that j - alpha is a unit. This implies explicit bounds on the number of these singular moduli.