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Mathematics > Number Theory

arXiv:1805.07167 (math)
[Submitted on 18 May 2018 (v1), last revised 7 Nov 2018 (this version, v2)]

Title:No singular modulus is a unit

Authors:Yu. Bilu, P. Habegger, L. Kühne
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Abstract: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.
Comments: Dedicated to David Masser on the occasion of his 70th birthday. Version 2 has a new title, updated references, and contains minor corrections. 31 pages, to appear in IMRN. Link to scripts this https URL
Subjects: Number Theory (math.NT)
Cite as: arXiv:1805.07167 [math.NT]
  (or arXiv:1805.07167v2 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1805.07167

Submission history

From: Philipp Habegger [view email]
[v1] Fri, 18 May 2018 12:24:13 UTC (27 KB)
[v2] Wed, 7 Nov 2018 12:07:54 UTC (28 KB)
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