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.
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Bruin, N., Dahmen, S.R. (2010). Visualizing Elements of Sha[3] in Genus 2 Jacobians. In: Hanrot, G., Morain, F., Thomé, E. (eds) Algorithmic Number Theory. ANTS 2010. Lecture Notes in Computer Science, vol 6197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14518-6_12
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DOI: https://doi.org/10.1007/978-3-642-14518-6_12
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