p-Adic Properties of Hauptmoduln with Applications to Moonshine
The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the j-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coeff...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2019 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2019
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210189 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | p-Adic Properties of Hauptmoduln with Applications to Moonshine / R.C. Chen, S. Marks, M. Tyler // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 42 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862716165076287488 |
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| author | Chen, R.C. Marks, S. Tyler, M. |
| author_facet | Chen, R.C. Marks, S. Tyler, M. |
| citation_txt | p-Adic Properties of Hauptmoduln with Applications to Moonshine / R.C. Chen, S. Marks, M. Tyler // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 42 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the j-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coefficients of the j-function satisfy congruences modulo pⁿ for p ∈ {2, 3, 5, 7, 11}, which led to the theory of p-adic modular forms. We combine these two aspects of the j-function to give a general theory of congruences modulo powers of primes satisfied by the Hauptmoduln appearing in monstrous moonshine. We prove that many of these Hauptmoduln satisfy such congruences, and we exhibit a relationship between these congruences and the group structure of the monster. We also find a distinguished class of subgroups of the monster with graded characters satisfying such congruences.
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| first_indexed | 2025-12-07T21:24:41Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210189 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:41Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chen, R.C. Marks, S. Tyler, M. 2025-12-03T14:31:45Z 2019 p-Adic Properties of Hauptmoduln with Applications to Moonshine / R.C. Chen, S. Marks, M. Tyler // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 42 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 11F11; 11F22; 11F33 arXiv: 1809.02913 https://nasplib.isofts.kiev.ua/handle/123456789/210189 https://doi.org/10.3842/SIGMA.2019.033 The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the j-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coefficients of the j-function satisfy congruences modulo pⁿ for p ∈ {2, 3, 5, 7, 11}, which led to the theory of p-adic modular forms. We combine these two aspects of the j-function to give a general theory of congruences modulo powers of primes satisfied by the Hauptmoduln appearing in monstrous moonshine. We prove that many of these Hauptmoduln satisfy such congruences, and we exhibit a relationship between these congruences and the group structure of the monster. We also find a distinguished class of subgroups of the monster with graded characters satisfying such congruences. We wish to thank Ken Ono, John Duncan, and Larry Rolen for their support and guidance, as well as Frank Calegari for useful information regarding lifts of characteristic p modular forms for the primes p = 2, 3. We thank our referees for helpful comments. We also thank Emory University, Princeton University, the Asa Griggs Candler Fund, and NSF grant DMS-1557960. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications p-Adic Properties of Hauptmoduln with Applications to Moonshine Article published earlier |
| spellingShingle | p-Adic Properties of Hauptmoduln with Applications to Moonshine Chen, R.C. Marks, S. Tyler, M. |
| title | p-Adic Properties of Hauptmoduln with Applications to Moonshine |
| title_full | p-Adic Properties of Hauptmoduln with Applications to Moonshine |
| title_fullStr | p-Adic Properties of Hauptmoduln with Applications to Moonshine |
| title_full_unstemmed | p-Adic Properties of Hauptmoduln with Applications to Moonshine |
| title_short | p-Adic Properties of Hauptmoduln with Applications to Moonshine |
| title_sort | p-adic properties of hauptmoduln with applications to moonshine |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210189 |
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