We give a parametrization of curves C of genus 2 with a maximal isotropic (ℤ/3)² in J[3], where J is the Jacobian variety of 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-3-1,
author = {Nils Bruin and E. Victor Flynn and Damiano Testa},
title = {Descent via (3,3)-isogeny on Jacobians of genus 2 curves},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {201-223},
zbl = {1311.11052},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-3-1}
}
Nils Bruin; E. Victor Flynn; Damiano Testa. Descent via (3,3)-isogeny on Jacobians of genus 2 curves. Acta Arithmetica, Tome 166 (2014) pp. 201-223. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-3-1/