Descent via $(3,3)$-isogeny on Jacobians of genus 2 curves
Volume 165 / 2014
Acta Arithmetica 165 (2014), 201-223
MSC: Primary 11G30; Secondary 11G10, 14H40.
DOI: 10.4064/aa165-3-1
Abstract
We give a parametrization of curves\nonbreakingspace $C$ of genus 2 with a maximal isotropic $({\mathbb Z}/3)^2$ in $J[3]$, where $J$ is the Jacobian variety of\nonbreakingspace $C$, and develop the theory required to perform descent via $(3,3)$-isogeny. We apply this to several examples, where it is shown that non-reducible Jacobians have non-trivial $3$-part of the Tate–Shafarevich group.
