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ANTS
2010
Springer

Visualizing Elements of Sha[3] in Genus 2 Jacobians

8 years 8 months ago
Visualizing Elements of Sha[3] in Genus 2 Jacobians
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 H1 (Gal(k/k), E) → H1 (Gal(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 ShafarevichTate 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.
Nils Bruin, Sander R. Dahmen
Added 15 Aug 2010
Updated 15 Aug 2010
Type Conference
Year 2010
Where ANTS
Authors Nils Bruin, Sander R. Dahmen
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