ARBITRARILY LARGE p-TORSION in TATE-SHAFAREVICH GROUPS

E. Victor Flynn; Ari Shnidman; Tom Fisher

doi:10.1017/S1474748024000392

ARBITRARILY LARGE p-TORSION in TATE-SHAFAREVICH GROUPS

E. Victor Flynn, Ari Shnidman, Tom Fisher

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that, for any prime p, there exist absolutely simple abelian varieties over Q with arbitrarily large p-Torsion in their Tate-Shafarevich groups. To prove this, we construct explicit up-covers of Jacobians of curves of the form yp = 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.

Original language	English
Journal	Journal of the Institute of Mathematics of Jussieu
DOIs	https://doi.org/10.1017/S1474748024000392
State	Accepted/In press - 2024

Bibliographical note

Publisher Copyright:
© The Author(s), 2024.

Keywords

abelian variety
Tate-Shafarevich group

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10.1017/S1474748024000392

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abstract = "We show that, for any prime p, there exist absolutely simple abelian varieties over Q with arbitrarily large p-Torsion in their Tate-Shafarevich groups. To prove this, we construct explicit up-covers of Jacobians of curves of the form yp = 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

Abstract

Bibliographical note

Keywords

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