Visualizing elements of Sha[3] in genus 2 jacobians

Abstract

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.

12 pages

Cited by in corpus (8)

• Isogeny volcanoes
• Cyclotomic matrices over the Eisenstein and Gaussian integers
• On the evaluation of modular polynomials
• Theta Lifts of Bianchi Modular Forms and Applications to Paramodularity
• On the computation of the HNF of a module over the ring of integers of a number field
• Failure of the Hasse principle for Enriques surfaces
• Geometric quadratic Chabauty and $p$-adic heights
• Small-span Hermitian matrices over quadratic integer rings