On the Tate-Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction. I.
type: Item
instance of: scholarly article
title: On the Tate-Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction. I.
zbMATH Open document ID: 1033.11028
author: Masato Kurihara
published in: Inventiones Mathematicae
publication date: 2003-08-17T00:00:00Z
review text: The Tate-Shafarevich group \(\text Ш(E,K)\) of an elliptic curve \(E\) over a number field \(K\) measures the obstruction that occurs if one wants to compute the rank \(r\) of \(E\) over \(K\). More precisely, one has an exact sequence \[ 0 \rightarrow E'(K)/\phi(E/K) \rightarrow S^{(\phi)}(E/K) \rightarrow \text Ш(E/K)[\phi] \rightarrow 0, \] where \(\phi: E/K \rightarrow E'/K\) is an isogeny of \(E\) over \(K\) to another elliptic curve \(E'\) over \(K\). Here, \(E'\) is \(E\) and \(\phi\) is essentially the multiplication by a prime \(p \in \mathbf P\). Whereas the Selmer group \(S^{(\phi)}(E/K)\) is computable, almost nothing is known about the Tate-Shafarevich group \(\text Ш(E/K)\). This group is conjectured to be finite, but even its subgroup \(\text Ш(E/K)[\phi]\) is ascertained to be finite only in some special cases. We denote the \(p\)-primary part of the Tate-Shafarevich group by \(\text Ш(E/K)[p]\). It is adaquate to apply Iwasawa theory to the elliptic curves \(E\) over say the rational field \(K=\mathbb{Q}\) and consider the asymptotic behaviour of \(\text Ш(E/K)[p]\) over the \(n\)th cyclotomic field \(K_n/\mathbb{Q}\) as \(n \to \infty\). Remember that \(K_n\) is a subfield of the cyclotomic \(\mathbb{Z}_p\)-extension \(K_{\infty}\) of \(\mathbb{Q}\). It has degree \[ [K_n:\mathbb{Q}] = p^n. \] If the \(p\)-primary part \(\text Ш(E/K_n)[p]\) is finite and has order \(p^{e_n}\), then there are numbers \(\lambda,\mu \in \mathbb{Z}_{\geq 0}\) and \(\nu \in \mathbb{Z}\) such that \[ e_n = \lambda n+\mu p^n+\nu \] for sufficiently large numbers \(n \in \mathbb{N}\) provided that the curve \(E\) has potentially ordinary reduction modulo \(p\). Of course, this is exactly the relation which holds for the \(p\)-class number of the field \(K_n\) and gives thus a justification for applying Iwasawa theory to elliptic curves. Iwasawa's result is not known for elliptic curves \(E\) that do not have potentially ordinary reduction modulo \(p\). (However, the author…
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11G07
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14L05
zbMATH DE Number: 1965424
zbMATH Keywords: Elliptic curve
zbMATH Keywords: Tate-Shafarevich group
zbMATH Keywords: Selmer group
zbMATH Keywords: Pontrjagin dual
zbMATH Keywords: formal group
zbMATH Keywords: Galois cohomology
zbMATH Keywords: cup product
zbMATH Keywords: de Rham cohomology
zbMATH Keywords: étale cohomology
zbMATH Keywords: crystalline cohomology
zbMATH Keywords: Néron period
zbMATH Keywords: Tate module
reviewed by: Horst G. Zimmer
MaRDI profile type: Publication
full work available at URL: s002220100206
OpenAlex ID: W2061678657
DOI: 10.1007/S002220100206
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description: scientific article; zbMATH DE number 1965424