A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curves
Abstract Let E be a Q‐isogeny class of elliptic curves defined over Q. The isogeny graph associated to E is a graph which has a vertex for each elliptic curve in the Q‐isogeny class E, and an edge for each cyclic Q‐isogeny of prime degree between elliptic curves in the isogeny class, with the degree...
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| Format: | Article |
| Language: | English |
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Wiley
2021-12-01
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| Series: | Transactions of the London Mathematical Society |
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| Online Access: | https://doi.org/10.1112/tlm3.12024 |
| _version_ | 1819176088480251904 |
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| author | Garen Chiloyan Álvaro Lozano‐Robledo |
| author_facet | Garen Chiloyan Álvaro Lozano‐Robledo |
| author_sort | Garen Chiloyan |
| collection | DOAJ |
| description | Abstract Let E be a Q‐isogeny class of elliptic curves defined over Q. The isogeny graph associated to E is a graph which has a vertex for each elliptic curve in the Q‐isogeny class E, and an edge for each cyclic Q‐isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny‐torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over Q of the corresponding elliptic curve. Then, the main result of the paper is a classification of all the possible isogeny‐torsion graphs that occur for Q‐isogeny classes of elliptic curves defined over the rationals. |
| first_indexed | 2024-12-22T21:05:12Z |
| format | Article |
| id | doaj.art-7b6a4b9f884f43a8bca7e92c4b623e32 |
| institution | Directory Open Access Journal |
| issn | 2052-4986 |
| language | English |
| last_indexed | 2024-12-22T21:05:12Z |
| publishDate | 2021-12-01 |
| publisher | Wiley |
| record_format | Article |
| series | Transactions of the London Mathematical Society |
| spelling | doaj.art-7b6a4b9f884f43a8bca7e92c4b623e322022-12-21T18:12:41ZengWileyTransactions of the London Mathematical Society2052-49862021-12-018113410.1112/tlm3.12024A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curvesGaren Chiloyan0Álvaro Lozano‐Robledo1Department of Mathematics University of Connecticut 341 Mansfield Road U1009 Storrs CT 06269 USADepartment of Mathematics University of Connecticut 341 Mansfield Road U1009 Storrs CT 06269 USAAbstract Let E be a Q‐isogeny class of elliptic curves defined over Q. The isogeny graph associated to E is a graph which has a vertex for each elliptic curve in the Q‐isogeny class E, and an edge for each cyclic Q‐isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny‐torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over Q of the corresponding elliptic curve. Then, the main result of the paper is a classification of all the possible isogeny‐torsion graphs that occur for Q‐isogeny classes of elliptic curves defined over the rationals.https://doi.org/10.1112/tlm3.1202411G05 (primary)14H52 (secondary) |
| spellingShingle | Garen Chiloyan Álvaro Lozano‐Robledo A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curves Transactions of the London Mathematical Society 11G05 (primary) 14H52 (secondary) |
| title | A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curves |
| title_full | A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curves |
| title_fullStr | A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curves |
| title_full_unstemmed | A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curves |
| title_short | A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curves |
| title_sort | classification of isogeny torsion graphs of q isogeny classes of elliptic curves |
| topic | 11G05 (primary) 14H52 (secondary) |
| url | https://doi.org/10.1112/tlm3.12024 |
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