Geodesic
Descent via $(3,3)$-isogeny on Jacobians of genus 2 curves
Nils Bruin1 ; E. Victor Flynn2 ; Damiano Testa3
1 Department of Mathematics Simon Fraser University Burnaby, BC, Canada V5A 1S6
2 Mathematical Institute University of Oxford 24–29 St. Giles Oxford OX1 3LB, United Kingdom
3 Mathematics Institute, Zeeman Building University of Warwick Coventry CV4 7AL, United Kingdom
Acta Arithmetica, Tome 165 (2014) no. 3, pp. 201-223 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/aa165-3-1
Keywords: parametrization curves nonbreakingspace genus maximal isotropic mathbb where jacobian variety nonbreakingspace develop theory required perform descent via isogeny apply several examples where shown non reducible jacobians have non trivial part tate shafarevich group

Nils Bruin 1 ; E. Victor Flynn 2 ; Damiano Testa 3

1 Department of Mathematics Simon Fraser University Burnaby, BC, Canada V5A 1S6
2 Mathematical Institute University of Oxford 24–29 St. Giles Oxford OX1 3LB, United Kingdom
3 Mathematics Institute, Zeeman Building University of Warwick Coventry CV4 7AL, United Kingdom 
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     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},
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Nils Bruin; E. Victor Flynn; Damiano Testa. Descent via $(3,3)$-isogeny on Jacobians of
 genus 2 curves. Acta Arithmetica, Tome 165 (2014) no. 3, pp. 201-223. doi: 10.4064/aa165-3-1

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