BILU, Yu.; HABEGGER, P.; KÜHNE, L.

doi:10.1093/imrn/rny274

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BILU, Yu.
Institut de Mathématiques de Bordeaux [IMB]

HABEGGER, P.

KÜHNE, L.

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International Mathematics Research Notices. 2020 n° 24, p. 10005-10041

Oxford University Press (OUP)

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A result of the second-named author states that there are only finitely many CM-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit. His proof depends on Duke's Equidistribution Theorem and is hence non-effective. 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 computations, 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 $\mathbb{C}^n$ not containing any special points.< Leer menos

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No singular modulus is a unit

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