STNB

STNB2025( 38a edició)

$3$-descent on genus $2$ Jacobians using visibility

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Lazar Radicevic

Resum

We show how to explicitly compute equations for everywhere locally soluble $3$-coverings of Jacobians of genus $2$ curves with a rational Weierstrass point, using the notion of visibility introduced by Cremona and Mazur. These $3$-coverings are abelian surface torsors, embedded in the projective space $\mathbb{P}^8$ as degree $18$ surfaces. They have points over every $p$-adic completion of $\mathbb{Q}$, but no rational points, and so are counterexamples to the Hasse principle and represent non-trivial elements of the Tate-Shafarevich group. Joint work in progress with Tom Fisher.

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