Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4
We work out a non-trivial example of lifting a so-called weak eigenform to a true, characteristic 0 eigenform. The weak eigenform is closely related to Ramanujan's tau function, whereas the characteristic 0 eigenform is attached to an elliptic curve defined over Q. We produce the lift by showin...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2018 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2018
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209515 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 / I. Kiming, N. Rustom // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862672532461584384 |
|---|---|
| author | Kiming, I. Rustom, N. |
| author_facet | Kiming, I. Rustom, N. |
| citation_txt | Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 / I. Kiming, N. Rustom // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We work out a non-trivial example of lifting a so-called weak eigenform to a true, characteristic 0 eigenform. The weak eigenform is closely related to Ramanujan's tau function, whereas the characteristic 0 eigenform is attached to an elliptic curve defined over Q. We produce the lift by showing that the coefficients of the initial, weak eigenform (almost all) occur as traces of Frobenii in the Galois representation on the 4-torsion of the elliptic curve. The example is remarkable as the initial form is known not to be liftable to any characteristic 0 eigenform of level 1. We use this example to illustrate certain questions that have arisen lately in the theory of modular forms modulo prime powers. We give a brief survey of those questions.
|
| first_indexed | 2025-12-07T15:35:50Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209515 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:35:50Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kiming, I. Rustom, N. 2025-11-24T10:07:17Z 2018 Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 / I. Kiming, N. Rustom // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 11F33; 11F80 arXiv: 1802.04976 https://nasplib.isofts.kiev.ua/handle/123456789/209515 https://doi.org/10.3842/SIGMA.2018.057 We work out a non-trivial example of lifting a so-called weak eigenform to a true, characteristic 0 eigenform. The weak eigenform is closely related to Ramanujan's tau function, whereas the characteristic 0 eigenform is attached to an elliptic curve defined over Q. We produce the lift by showing that the coefficients of the initial, weak eigenform (almost all) occur as traces of Frobenii in the Galois representation on the 4-torsion of the elliptic curve. The example is remarkable as the initial form is known not to be liftable to any characteristic 0 eigenform of level 1. We use this example to illustrate certain questions that have arisen lately in the theory of modular forms modulo prime powers. We give a brief survey of those questions. The authors would like to thank Shaunak Deo and Gabor Wiese for interesting discussions relating to this paper, as well as to other questions concerning modular forms modulo prime powers. We thank Ariel Pacetti for comments on the first draft of the paper. We also thank the anonymous referees for comments and suggestions that helped improve the exposition. The second author was supported by a Postdoctoral Fellowship at the National Center for Theoretical Sciences, Taipei, Taiwan. The first author would like to thank Noriko Yui for good contact, collaboration, and interesting exchange over many years. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 Article published earlier |
| spellingShingle | Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 Kiming, I. Rustom, N. |
| title | Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 |
| title_full | Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 |
| title_fullStr | Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 |
| title_full_unstemmed | Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 |
| title_short | Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 |
| title_sort | dihedral group, 4-torsion on an elliptic curve, and a peculiar eigenform modulo 4 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209515 |
| work_keys_str_mv | AT kimingi dihedralgroup4torsiononanellipticcurveandapeculiareigenformmodulo4 AT rustomn dihedralgroup4torsiononanellipticcurveandapeculiareigenformmodulo4 |