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.

• Type: Conference
• Year: 2010
• Where: ANTS
• Authors: Nils Bruin, Sander R. Dahmen
• Keywords: Abelian Surface, Algorithms, ANTS 2010, Elliptic Curve, Number field