Kolyvagin’s result on the vanishing of $\protect \Sha(E/K)[p^\infty ]$ and its consequences for anticyclotomic Iwasawa theory

Ahmed Matar; Jan Nekovář

doi:10.5802/jtnb.1091

Kolyvagin’s result on the vanishing of

Ш (E / K) [p^{\infty}]

and its consequences for anticyclotomic Iwasawa theory

Ahmed Matar ¹; Jan Nekovář ²

¹ Department of Mathematics University of Bahrain P.O. Box 32038 Sukhair, Bahrain
² Sorbonne Université Campus Pierre et Marie Curie Institut de Mathématiques de Jussieu Théorie des Nombres, Case 247 4 place Jussieu 75252 Paris cedex 05, France

Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 455-501.

corrected-by Correction to the paper “Kolyvagin’s result on the vanishing of

Ш (E / K) [p^{\infty}]

and its consequences for anticyclotomic Iwasawa theory”

Abstract

We discuss improvements of Kolyvagin’s classical result about the vanishing of the $p$ -primary part of the Tate–Šafarevič group of an elliptic curve $E$ (defined over $ℚ$ ) over an imaginary quadratic field $K$ satisfying the Heegner hypothesis for which the basic Heegner point $y_{K} \in E (K)$ is not divisible by an odd prime $p$ . Combining Kolyvagin’s theorem with a new abstract Iwasawa-theoretical result, we deduce, under suitable assumptions, that similar vanishing holds for all layers in the anticyclotomic $Z_{p}$ -extension of $K$ .

Received: 2018-11-19
Accepted: 2019-07-14
Published online: 2019-10-29

DOI: 10.5802/jtnb.1091

Classification: 11G05, 11G18, 11G40, 14G10, 14G35
Keywords: Heegner points, elliptic curves, Iwasawa theory

Author's affiliations:

Ahmed Matar ¹; Jan Nekovář ²

¹ Department of Mathematics University of Bahrain P.O. Box 32038 Sukhair, Bahrain
² Sorbonne Université Campus Pierre et Marie Curie Institut de Mathématiques de Jussieu Théorie des Nombres, Case 247 4 place Jussieu 75252 Paris cedex 05, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Kolyvagin{\textquoteright}s result on the vanishing of $\protect \Sha(E/K)[p^\infty ]$ and its consequences for anticyclotomic {Iwasawa} theory},
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Ahmed Matar; Jan Nekovář. Kolyvagin’s result on the vanishing of $\protect \Sha(E/K)[p^\infty ]$ and its consequences for anticyclotomic Iwasawa theory. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 455-501. doi : 10.5802/jtnb.1091. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1091/

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