Realization of Groups with Pairing as Jacobians of Finite Graphs
Abstract We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yiel...
| Main Authors: | , , , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer International Publishing
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131386 |
| _version_ | 1811095252385136640 |
|---|---|
| author | Gaudet, Louis Jensen, David Ranganathan, Dhruv Wawrykow, Nicholas Weisman, Theodore |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Gaudet, Louis Jensen, David Ranganathan, Dhruv Wawrykow, Nicholas Weisman, Theodore |
| author_sort | Gaudet, Louis |
| collection | MIT |
| description | Abstract
We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simple graph. There exist infinite families of finite abelian groups that do not occur as the Jacobians of simple graphs. |
| first_indexed | 2024-09-23T16:14:03Z |
| format | Article |
| id | mit-1721.1/131386 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:14:03Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1313862023-01-11T17:30:28Z Realization of Groups with Pairing as Jacobians of Finite Graphs Gaudet, Louis Jensen, David Ranganathan, Dhruv Wawrykow, Nicholas Weisman, Theodore Massachusetts Institute of Technology. Department of Mathematics Abstract We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simple graph. There exist infinite families of finite abelian groups that do not occur as the Jacobians of simple graphs. 2021-09-20T17:16:52Z 2021-09-20T17:16:52Z 2018-11-02 2020-09-24T21:10:21Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131386 en https://doi.org/10.1007/s00026-018-0406-0 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Nature Switzerland AG application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Gaudet, Louis Jensen, David Ranganathan, Dhruv Wawrykow, Nicholas Weisman, Theodore Realization of Groups with Pairing as Jacobians of Finite Graphs |
| title | Realization of Groups with Pairing as Jacobians of Finite Graphs |
| title_full | Realization of Groups with Pairing as Jacobians of Finite Graphs |
| title_fullStr | Realization of Groups with Pairing as Jacobians of Finite Graphs |
| title_full_unstemmed | Realization of Groups with Pairing as Jacobians of Finite Graphs |
| title_short | Realization of Groups with Pairing as Jacobians of Finite Graphs |
| title_sort | realization of groups with pairing as jacobians of finite graphs |
| url | https://hdl.handle.net/1721.1/131386 |
| work_keys_str_mv | AT gaudetlouis realizationofgroupswithpairingasjacobiansoffinitegraphs AT jensendavid realizationofgroupswithpairingasjacobiansoffinitegraphs AT ranganathandhruv realizationofgroupswithpairingasjacobiansoffinitegraphs AT wawrykownicholas realizationofgroupswithpairingasjacobiansoffinitegraphs AT weismantheodore realizationofgroupswithpairingasjacobiansoffinitegraphs |