Details
| Original language | English |
|---|---|
| Pages (from-to) | 10005-10041 |
| Number of pages | 37 |
| Journal | International Mathematics Research Notices |
| Volume | 2020 |
| Issue number | 24 |
| Early online date | 6 Dec 2018 |
| Publication status | Published - 1 Dec 2020 |
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.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: International Mathematics Research Notices, Vol. 2020, No. 24, 01.12.2020, p. 10005-10041.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - No Singular Modulus Is a Unit
AU - Bilu, Yuri
AU - Habegger, Philipp
AU - Kühne, Lars
N1 - Funding Information: This work was supported by the University of Basel [to Yu.B.]; Fields Institute [Thematic Program on Unlikely Intersections, Heights, and Efficient Congruencing to Yu.B. and L.K.]; Xiamen University [to Yu.B.]; Max-Planck Institute for Mathematics [to L.K.]; and Swiss National Science Foundation [168055 to L.K.].
PY - 2020/12/1
Y1 - 2020/12/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85089023807&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1805.07167
DO - 10.48550/arXiv.1805.07167
M3 - Article
AN - SCOPUS:85089023807
VL - 2020
SP - 10005
EP - 10041
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
SN - 1073-7928
IS - 24
ER -