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Reference: Bruin, N and Flynn, EV, (2006). Exhibiting SHA[2] on hyperelliptic Jacobians. Journal of Number Theory, 118 (2), 266-291.Citable link to this page:

 

Exhibiting SHA[2] on hyperelliptic Jacobians

Abstract: We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of genus 2, we also demonstrate a connection with degree 4 del Pezzo surfaces, and show how the Brauer-Manin obstruction on these surfaces can be used to compute members of the Shafarevich-Tate group of Jacobians. We derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the Shafarevich-Tate group. Finally, we prove that under certain conditions, the visualisation dimension for order 2 cocycles of Jacobians of certain genus 2 curves is 4 rather than the general bound of 32. © 2005 Elsevier Inc. All rights reserved.

Peer Review status:Peer reviewedPublication status:PublishedVersion:Publisher versionNotes:Copyright 2005 Elsevier B.V. All rights reserved. Re-use of this article is permitted in accordance with the Terms and Conditions set out at http://www.elsevier.com/open-access/userlicense/1.0/

Bibliographic Details

Publisher: Elsevier

Publisher Website: http://www.elsevier.com/

Journal: Journal of Number Theorysee more from them

Publication Website: http://www.sciencedirect.com/science/journal/0022314X

Issue Date: 2006-6

pages:266-291Identifiers

Urn: uuid:2f8e607e-4f8a-4e22-9ff0-3a1b215be8ae

Source identifier: 148108

Doi: https://doi.org/10.1016/j.jnt.2005.10.007

Issn: 0022-314X Item Description

Type: Journal article;

Language: eng

Version: Publisher versionKeywords: Brauer-Manin obstruction Higher genus curves Jacobians Shafarevich-Tate group Visualisation Tiny URL: pubs:148108

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Autor: Bruin, N - - - Flynn, EV - institutionUniversity of Oxford Oxford, MPLS, Mathematical Institute grantNumberGR-R82975-01 fundingEn

Fuente: https://ora.ox.ac.uk/objects/uuid:2f8e607e-4f8a-4e22-9ff0-3a1b215be8ae



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