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  Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes

Radičević, L. (2022). Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes. Research in Number Theory, 8(4): 72. doi:10.1007/s40993-022-00374-1.

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Item Permalink: https://hdl.handle.net/21.11116/0000-000B-3647-9 Version Permalink: https://hdl.handle.net/21.11116/0000-0011-2A5D-7
Genre: Journal Article

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© The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0
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 Creators:
Radičević, Lazar1, Author                 
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Number Theory
 Abstract: We consider the Kolyvagin cohomology classes associated to an elliptic curve
$E$ defined over $\mathbb{Q}$ from a computational point of view. We explain
how to go from a model of a class as an element of
$(E(L)/pE(L))^{\mathrm{Gal}(L/\mathbb{Q})}$, where $p$ is prime and $L$ is a
dihedral extension of $\mathbb{Q}$ of degree $2p$, to a geometric model as a
genus one curve embedded in $\mathbb{P}^{p-1}$. We adapt the existing methods
to compute Heegner points to our situation, and explicitly compute them as
elements of $E(L)$. Finally, we compute explicit equations for several genus
one curves that represent non-trivial elements of the p-torsion part of the
Tate-Shafarevich group of $E$, for $p \leq 11$, and hence are counterexamples
to the Hasse principle.

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Language(s): eng - English
 Dates: 2022
 Publication Status: Published online
 Pages: 25
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 2112.02016
DOI: 10.1007/s40993-022-00374-1
 Degree: -

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Title: Research in Number Theory
Source Genre: Journal
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Publ. Info: Springer
Pages: - Volume / Issue: 8 (4) Sequence Number: 72 Start / End Page: - Identifier: -