For fixed non-CM α, only finitely many CM j satisfy that j-α is an algebraic unit, with explicit bounds given.
Lehmer: Properties of the coefficients of the modular invariant J(τ)
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Integrality properties in the Moduli Space of Elliptic Curves: CM Case
For fixed non-CM α, only finitely many CM j satisfy that j-α is an algebraic unit, with explicit bounds given.