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Visualizing Elements of Sha[3] in Genus 2 Jacobians

  • Nils Bruin
  • Sander R. Dahmen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)

Abstract

Mazur proved that any element ξ of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that ξ lies in the kernel of the natural homomorphism between the cohomology groups \(H^1({\rm Gal}({\overline k}/k),E) \rightarrow H^1({\rm Gal}({\overline k}/k),A)\). However, the abelian surface in Mazur’s construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.

Keywords

Elliptic Curve Elliptic Curf Abelian Variety Abelian Surface Galois Cohomology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Nils Bruin
    • 1
  • Sander R. Dahmen
    • 1
  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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