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No singular modulus is a unit

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorBILU, Yu.
dc.contributor.authorHABEGGER, P.
dc.contributor.authorKÜHNE, L.
dc.date.accessioned2024-04-04T03:04:31Z
dc.date.available2024-04-04T03:04:31Z
dc.date.issued2020
dc.identifier.issn1073-7928
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/193161
dc.description.abstractEnA 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.
dc.language.isoen
dc.publisherOxford University Press (OUP)
dc.title.enNo singular modulus is a unit
dc.typeArticle de revue
dc.identifier.doi10.1093/imrn/rny274
dc.subject.halMathématiques [math]/Théorie des nombres [math.NT]
dc.identifier.arxiv1805.07167
bordeaux.journalInternational Mathematics Research Notices
bordeaux.page10005-10041
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue24
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-01914592
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01914592v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=International%20Mathematics%20Research%20Notices&rft.date=2020&rft.issue=24&rft.spage=10005-10041&rft.epage=10005-10041&rft.eissn=1073-7928&rft.issn=1073-7928&rft.au=BILU,%20Yu.&HABEGGER,%20P.&K%C3%9CHNE,%20L.&rft.genre=article


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