Supersingular Elliptic Curves and Moonshine
We generalize a theorem of Ogg on supersingular j-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous character...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2019 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210068 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Supersingular Elliptic Curves and Moonshine / V.M. Aricheta // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 32 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862751965647208448 |
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| author | Aricheta, V.M. |
| author_facet | Aricheta, V.M. |
| citation_txt | Supersingular Elliptic Curves and Moonshine / V.M. Aricheta // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We generalize a theorem of Ogg on supersingular j-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. We also present a connection between supersingular elliptic curves and umbral moonshine. Finally, we present a procedure for explicitly computing invariants of supersingular elliptic curves with level structure.
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| first_indexed | 2025-12-07T21:24:16Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210068 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:16Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Aricheta, V.M. 2025-12-02T09:37:25Z 2019 Supersingular Elliptic Curves and Moonshine / V.M. Aricheta // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H52; 11F06; 11F11; 11F22; 11F37; 20D08 arXiv: 1809.07421 https://nasplib.isofts.kiev.ua/handle/123456789/210068 https://doi.org/10.3842/SIGMA.2019.007 We generalize a theorem of Ogg on supersingular j-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. We also present a connection between supersingular elliptic curves and umbral moonshine. Finally, we present a procedure for explicitly computing invariants of supersingular elliptic curves with level structure. We are immensely indebted to John Duncan for his constant advice and encouragement throughout the project, without which this work would not have been possible. We are also grateful to Luca Candelori, Scott Carnahan, Simon Norton, Preston Wake, Robert Wilson, and David Zureick-Brown for taking the time to answer our questions, and to Ken Ono for posing the original problem, which motivated this research. We thank the referees for the helpful suggestions and comments. We wrote part of this paper during our stay at the Erwin Schr¨odinger International Institute for Mathematics and Physics (ESI) from 10 to 14 September 2018; we are thankful to ESI for their support. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Supersingular Elliptic Curves and Moonshine Article published earlier |
| spellingShingle | Supersingular Elliptic Curves and Moonshine Aricheta, V.M. |
| title | Supersingular Elliptic Curves and Moonshine |
| title_full | Supersingular Elliptic Curves and Moonshine |
| title_fullStr | Supersingular Elliptic Curves and Moonshine |
| title_full_unstemmed | Supersingular Elliptic Curves and Moonshine |
| title_short | Supersingular Elliptic Curves and Moonshine |
| title_sort | supersingular elliptic curves and moonshine |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210068 |
| work_keys_str_mv | AT arichetavm supersingularellipticcurvesandmoonshine |