Genus two curves with full $$\sqrt{3}$$-level structure and Tate–Shafarevich groups

Bruin, Nils; Flynn, E. V.; Ari, Shnidman,

doi:10.1007/s00029-023-00839-w

Sel. Math. New Ser.

2023

DOI: 10.1007/s00029-023-00839-w

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Genus two curves with full $$\sqrt{3}$$-level structure and Tate–Shafarevich groups

Nils Bruin

E. V. Flynn

Shnidman, Ari

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2024

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ARBITRARILY LARGE p-TORSION IN TATE-SHAFAREVICH GROUPS

Flynn,

Shnidman,

Fisher

2024

J. Inst. Math. Jussieu

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We show that, for any prime p, there exist absolutely simple abelian varieties over $\mathbb {Q}$ with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit $\mu _p$ -covers of Jacobians of curves of the form $y^p = x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.

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ARBITRARILY LARGE p-TORSION IN TATE-SHAFAREVICH GROUPS

Flynn,

Shnidman,

Fisher

2024

J. Inst. Math. Jussieu

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show abstract

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Genus two curves with full $$\sqrt{3}$$-level structure and Tate–Shafarevich groups

Cited by 1 publication

References 18 publications

ARBITRARILY LARGE p-TORSION IN TATE-SHAFAREVICH GROUPS

ARBITRARILY LARGE p-TORSION IN TATE-SHAFAREVICH GROUPS

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