Visualizing elements of Sha[3] in genus 2 jacobians

Details Visualizing elements of Sha[3] in genus 2 jacobians Visualizing elements of Sha[3] in genus 2 jacobians

2010-01-29

arxiv.org/abs/1001.5302v1

Algorithmic Number Theory Lecture Notes in Computer Science, 2010, Volume 6197/2010, 110-125

Mathematics

Number Theory

12 pages

Scientific paper

10.1007/978-3-642-14518-6

Mazur proved that any element xi 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 xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(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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