On the 2-Selmer group of Jacobians of hyperelliptic curves

Barrera Salazar, Daniel - Pacetti, Ariel - Tornaría, Gonzalo

Resumen:

Let C be a hyperelliptic curve y2=p(x) defined over a number field K with p(x) integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the 2-Selmer group of the Jacobian of C in terms of the class group of the K-algebra K[x]/(p(x)). Our main result is a formula relating these two quantities under some mild hypothesis. We provide some examples that prove that our lower and upper bounds are as sharp as possible. As a first application, we study the rank distribution of the 2-Selmer group in families of quadratic twists. Under some extra hypothesis we prove that among prime quadratic twists, a positive proportion has fixed 2-Selmer group. As a second application, we study the family of octic twists of the genus 2 curve y2=x5+x.


Detalles Bibliográficos
2023
MATHEMATICS - NUMBER THEORY
Inglés
Universidad de la República
COLIBRI
https://hdl.handle.net/20.500.12008/45028
Acceso abierto
Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)
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author Barrera Salazar, Daniel
author2 Pacetti, Ariel
Tornaría, Gonzalo
author2_role author
author
author_facet Barrera Salazar, Daniel
Pacetti, Ariel
Tornaría, Gonzalo
author_role author
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collection COLIBRI
dc.contributor.filiacion.none.fl_str_mv Barrera Salazar Daniel
Pacetti Ariel
Tornaría Gonzalo, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.
dc.creator.none.fl_str_mv Barrera Salazar, Daniel
Pacetti, Ariel
Tornaría, Gonzalo
dc.date.accessioned.none.fl_str_mv 2024-08-01T12:32:51Z
dc.date.available.none.fl_str_mv 2024-08-01T12:32:51Z
dc.date.issued.none.fl_str_mv 2023
dc.description.abstract.none.fl_txt_mv Let C be a hyperelliptic curve y2=p(x) defined over a number field K with p(x) integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the 2-Selmer group of the Jacobian of C in terms of the class group of the K-algebra K[x]/(p(x)). Our main result is a formula relating these two quantities under some mild hypothesis. We provide some examples that prove that our lower and upper bounds are as sharp as possible. As a first application, we study the rank distribution of the 2-Selmer group in families of quadratic twists. Under some extra hypothesis we prove that among prime quadratic twists, a positive proportion has fixed 2-Selmer group. As a second application, we study the family of octic twists of the genus 2 curve y2=x5+x.
dc.description.es.fl_txt_mv Versión permitida preprint.
dc.format.extent.es.fl_str_mv 28 h.
dc.format.mimetype.es.fl_str_mv application/pdf
dc.identifier.citation.es.fl_str_mv Barrera Salazar, D, Pacetti, A y Tornaría, G. "On the 2-Selmer group of Jacobians of hyperelliptic curves" [Preprint]. Publicado en: Mathematics (Number Theory). 2023, arXiv:2308.08663, ago. 2023, pp.1-27. DOI: 10.48550/arXiv.2308.08663
dc.identifier.doi.none.fl_str_mv 10.48550/arXiv.2308.08663
dc.identifier.uri.none.fl_str_mv https://hdl.handle.net/20.500.12008/45028
dc.language.iso.none.fl_str_mv en
eng
dc.publisher.es.fl_str_mv arXiv
dc.relation.ispartof.es.fl_str_mv Mathematics (Number Theory), arXiv:2308.08663, ago. 2023, pp.1-27
dc.rights.license.none.fl_str_mv Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
dc.source.none.fl_str_mv reponame:COLIBRI
instname:Universidad de la República
instacron:Universidad de la República
dc.subject.other.es.fl_str_mv MATHEMATICS - NUMBER THEORY
dc.title.none.fl_str_mv On the 2-Selmer group of Jacobians of hyperelliptic curves
dc.type.es.fl_str_mv Preprint
dc.type.none.fl_str_mv info:eu-repo/semantics/preprint
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description Versión permitida preprint.
eu_rights_str_mv openAccess
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identifier_str_mv Barrera Salazar, D, Pacetti, A y Tornaría, G. "On the 2-Selmer group of Jacobians of hyperelliptic curves" [Preprint]. Publicado en: Mathematics (Number Theory). 2023, arXiv:2308.08663, ago. 2023, pp.1-27. DOI: 10.48550/arXiv.2308.08663
10.48550/arXiv.2308.08663
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repository.name.fl_str_mv COLIBRI - Universidad de la República
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rights_invalid_str_mv Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)
spelling Barrera Salazar DanielPacetti ArielTornaría Gonzalo, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.2024-08-01T12:32:51Z2024-08-01T12:32:51Z2023Barrera Salazar, D, Pacetti, A y Tornaría, G. "On the 2-Selmer group of Jacobians of hyperelliptic curves" [Preprint]. Publicado en: Mathematics (Number Theory). 2023, arXiv:2308.08663, ago. 2023, pp.1-27. DOI: 10.48550/arXiv.2308.08663https://hdl.handle.net/20.500.12008/4502810.48550/arXiv.2308.08663Versión permitida preprint.Let C be a hyperelliptic curve y2=p(x) defined over a number field K with p(x) integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the 2-Selmer group of the Jacobian of C in terms of the class group of the K-algebra K[x]/(p(x)). Our main result is a formula relating these two quantities under some mild hypothesis. We provide some examples that prove that our lower and upper bounds are as sharp as possible. As a first application, we study the rank distribution of the 2-Selmer group in families of quadratic twists. Under some extra hypothesis we prove that among prime quadratic twists, a positive proportion has fixed 2-Selmer group. As a second application, we study the family of octic twists of the genus 2 curve y2=x5+x.Submitted by Egaña Florencia (florega@gmail.com) on 2024-07-31T19:14:13Z No. of bitstreams: 2 license_rdf: 25790 bytes, checksum: 489f03e71d39068f329bdec8798bce58 (MD5) 2308.08663v1.pdf: 645309 bytes, checksum: 78908cdcda64faa21e8ac5f3ae6080dc (MD5)Approved for entry into archive by Faget Cecilia (lfaget@fcien.edu.uy) on 2024-08-01T11:32:51Z (GMT) No. of bitstreams: 2 license_rdf: 25790 bytes, checksum: 489f03e71d39068f329bdec8798bce58 (MD5) 2308.08663v1.pdf: 645309 bytes, checksum: 78908cdcda64faa21e8ac5f3ae6080dc (MD5)Made available in DSpace by Luna Fabiana (fabiana.luna@seciu.edu.uy) on 2024-08-01T12:32:51Z (GMT). No. of bitstreams: 2 license_rdf: 25790 bytes, checksum: 489f03e71d39068f329bdec8798bce58 (MD5) 2308.08663v1.pdf: 645309 bytes, checksum: 78908cdcda64faa21e8ac5f3ae6080dc (MD5) Previous issue date: 202328 h.application/pdfenengarXivMathematics (Number Theory), arXiv:2308.08663, ago. 2023, pp.1-27Las obras depositadas en el Repositorio se rigen por la Ordenanza de los Derechos de la Propiedad Intelectual de la Universidad de la República.(Res. Nº 91 de C.D.C. de 8/III/1994 – D.O. 7/IV/1994) y por la Ordenanza del Repositorio Abierto de la Universidad de la República (Res. Nº 16 de C.D.C. de 07/10/2014)info:eu-repo/semantics/openAccessLicencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)MATHEMATICS - NUMBER THEORYOn the 2-Selmer group of Jacobians of hyperelliptic curvesPreprintinfo:eu-repo/semantics/preprintinfo:eu-repo/semantics/submittedVersionreponame:COLIBRIinstname:Universidad de la Repúblicainstacron:Universidad de la RepúblicaBarrera Salazar, DanielPacetti, ArielTornaría, GonzaloLICENSElicense.txtlicense.txttext/plain; charset=utf-84267http://localhost:8080/xmlui/bitstream/20.500.12008/45028/5/license.txt6429389a7df7277b72b7924fdc7d47a9MD55CC-LICENSElicense_urllicense_urltext/plain; charset=utf-850http://localhost:8080/xmlui/bitstream/20.500.12008/45028/2/license_urla006180e3f5b2ad0b88185d14284c0e0MD52license_textlicense_texttext/html; charset=utf-822527http://localhost:8080/xmlui/bitstream/20.500.12008/45028/3/license_textdf0749cf944f9d2754bc76e8ce56250cMD53license_rdflicense_rdfapplication/rdf+xml; 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- Universidad de la Repúblicafalse
spellingShingle On the 2-Selmer group of Jacobians of hyperelliptic curves
Barrera Salazar, Daniel
MATHEMATICS - NUMBER THEORY
status_str submittedVersion
title On the 2-Selmer group of Jacobians of hyperelliptic curves
title_full On the 2-Selmer group of Jacobians of hyperelliptic curves
title_fullStr On the 2-Selmer group of Jacobians of hyperelliptic curves
title_full_unstemmed On the 2-Selmer group of Jacobians of hyperelliptic curves
title_short On the 2-Selmer group of Jacobians of hyperelliptic curves
title_sort On the 2-Selmer group of Jacobians of hyperelliptic curves
topic MATHEMATICS - NUMBER THEORY
url https://hdl.handle.net/20.500.12008/45028

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