No Singular Modulus Is a Unit

authored by

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Universite de Bordeaux
University of Basel

Type

Article

Journal

International Mathematics Research Notices

Volume

2020

Pages

10005-10041

No. of pages

37

ISSN

1073-7928

Publication date

01.12.2020

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Mathematics(all)

Electronic version(s)

https://doi.org/10.48550/arXiv.1805.07167 (Access: Open)
https://doi.org/10.1093/imrn/rny274 (Access: Closed)