For each algebraic number α ∈ℚ̄, 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 − α 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 α is not a singular modulus itself. We give an explicit bound, which depends only on α, on the discriminant Δ associated with a singular modulus j such that j − α is a unit. This implies explicit bounds on the number of these singular moduli.

中文翻译：

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椭圆曲线模空间中的积分性质：CM情况

对于每个代数数α ∈ℚ̄, Habegger [P. Habegger，作为代数单位的奇异模量，代数数论 9(7) (2015) 1515–1524] 表明只有有限多个奇异模j这样j - α是一个代数单位。他的结果使用了杜克的平均分布定理，因此无效。在本文中，我们给出了 Habegger 结果的有效证明，假设α本身不是奇异模数。我们给出一个明确的界限，它只取决于α, 在判别式上Δ与奇异模数相关j这样j - α是一个单位。这意味着对这些奇异模数的数量有明确的限制。