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Covering collections and a challenge problem of Serre

E. Victor Flynn; Joseph L. Wetherell — 2001

Acta Arithmetica

Non-trivial $Ш$ in the Jacobian of an infinite family of curves of genus 2

Anna Arnth-Jensen; E. Victor Flynn — 2009

Journal de Théorie des Nombres de Bordeaux

We give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an associated family of surfaces which violate the Hasse principle.

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

Nils Bruin; E. Victor Flynn; Damiano Testa — 2014

Acta Arithmetica

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.

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Currently displaying 1 – 3 of 3

Covering collections and a challenge problem of Serre

Non-trivial Ш in the Jacobian of an infinite family of curves of genus 2

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

Non-trivial $Ш$ in the Jacobian of an infinite family of curves of genus 2