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)


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.


Elliptic Curve Elliptic Curf Abelian Variety Abelian Surface Galois Cohomology 
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© 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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