No Singular Modulus Is a Unit

verfasst von

Yuri Bilu, Philipp Habegger, Lars Kühne

Abstract

A result of the 2nd-named author states that there are only finitely many complex multiplication (CM)-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit. His proof depends on Duke's equidistribution theorem and is hence noneffective. 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 arguments, 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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Externe Organisation(en)

Universite de Bordeaux
Universität Basel

Typ

Artikel

Journal

International Mathematics Research Notices

Band

2020

Seiten

10005-10041

Anzahl der Seiten

37

ISSN

1073-7928

Publikationsdatum

01.12.2020

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Mathematik (insg.)

Elektronische Version(en)

https://doi.org/10.48550/arXiv.1805.07167 (Zugang: Offen)
https://doi.org/10.1093/imrn/rny274 (Zugang: Geschlossen)