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

Anna Arnth-JensenE. 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.

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