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The least degree of a CM point on a modular curve

2022; Wiley; Volume: 105; Issue: 2 Linguagem: Inglês

10.1112/jlms.12518

1469-7750

Pete L. Clark, Tyler Genao, Paul Pollack, Frederick Saia,

Coding theory and cryptography

Journal of the London Mathematical SocietyVolume 105, Issue 2 p. 825-883 RESEARCH ARTICLE The least degree of a CM point on a modular curve Pete L. Clark, Corresponding Author Pete L. Clark [email protected] Department of Mathematics, University of Georgia, Athens, Georgia, USA Correspondence Pete L. Clark, Department of Mathematics, University of Georgia, Athens, GA 30602. Email: [email protected]Search for more papers by this authorTyler Genao, Tyler Genao Department of Mathematics, University of Georgia, Athens, Georgia, USASearch for more papers by this authorPaul Pollack, Paul Pollack Department of Mathematics, University of Georgia, Athens, Georgia, USASearch for more papers by this authorFrederick Saia, Frederick Saia Department of Mathematics, University of Georgia, Athens, Georgia, USASearch for more papers by this author Pete L. Clark, Corresponding Author Pete L. Clark [email protected] Department of Mathematics, University of Georgia, Athens, Georgia, USA Correspondence Pete L. Clark, Department of Mathematics, University of Georgia, Athens, GA 30602. Email: [email protected]Search for more papers by this authorTyler Genao, Tyler Genao Department of Mathematics, University of Georgia, Athens, Georgia, USASearch for more papers by this authorPaul Pollack, Paul Pollack Department of Mathematics, University of Georgia, Athens, Georgia, USASearch for more papers by this authorFrederick Saia, Frederick Saia Department of Mathematics, University of Georgia, Athens, Georgia, USASearch for more papers by this author First published: 11 February 2022 https://doi.org/10.1112/jlms.12518Citations: 3Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract For a modular curve X = X 0 ( N ) $X = X_0(N)$ , X 1 ( N ) $X_1(N)$ or X 1 ( M , N ) $X_1(M,N)$ defined over Q $\mathbb {Q}$ , we denote by d CM ( X ) $d_{\operatorname{CM}}(X)$ the least degree of a CM point on X $X$ . For each discriminant Δ < 0 $\Delta < 0$ , we determine the least degree of a point on X 0 ( N ) $X_0(N)$ with CM by the order of discriminant Δ $\Delta$ . This places us in a position to study d CM ( X ) $d_{\operatorname{CM}}(X)$ as an 'arithmetic function' and we do so, obtaining various upper bounds, lower bounds and typical bounds. We deduce that all but finitely many curves in each of the families have sporadic CM points. 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