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Descent via $(3,3)$-isogeny on Jacobians of genus 2 curves

Volume 165 / 2014

Nils Bruin, E. Victor Flynn, Damiano Testa 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.

Authors

  • Nils BruinDepartment of Mathematics
    Simon Fraser University
    Burnaby, BC, Canada V5A 1S6
    e-mail
  • E. Victor FlynnMathematical Institute
    University of Oxford
    24–29 St. Giles
    Oxford OX1 3LB, United Kingdom
    e-mail
  • Damiano TestaMathematics Institute, Zeeman Building
    University of Warwick
    Coventry CV4 7AL, United Kingdom
    e-mail

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