The Šafarevič-Tate group in cyclotomic \(\mathbb Z_p\)-extensions at supersingular primes
type: Item
instance of: scholarly article
zbMATH Open document ID: 1288.11060
author: Florian Sprung
publication date: 2013-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.1936
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 6211854
zbMATH Keywords: Coleman map
zbMATH Keywords: Tate-Shafarevich group
zbMATH Keywords: \({\mathbb Z}_p\)-extension
zbMATH Keywords: supersingular prime
zbMATH Keywords: good supersingular reduction
zbMATH Keywords: elliptic curve
zbMATH Keywords: supersingular reduction
zbMATH Keywords: Selmer group
zbMATH Keywords: Iwasawa theory
MaRDI profile type: Publication
OpenAlex ID: W2313921919
arXiv ID: 1106.1936
DOI: 10.1515/CRELLE-2012-0031
title: The Šafarevič-Tate group in cyclotomic \(\mathbb Z_p\)-extensions at supersingular primes
published in: Journal für die Reine und Angewandte Mathematik
review text: The main goal of this paper is the study of the asymptotic growth of the \(p\)--primary component of the Shafarevich-Tate group in the cyclotomic direction at any odd prime of good supersingular reduction. Not much is known about its size and how it behaves according to the number field. \textit{B. Mazur} [Invent. Math. 18, 183--266 (1972; Zbl 0245.14015)] studied the case of a \({\mathbb Z}_p\)-extension when \(p\) has good ordinary reduction. That is, let \(E\) be an elliptic curve over \({\mathbb Q}\) and \(p\) an odd prime of good reduction and let \({\mathbb Q}_{\infty}\) be the cyclotomic \({\mathbb Z}_p\)-extension, \(\Gamma:=\mathrm{Gal}({\mathbb Q}_{\infty}/{\mathbb Q}) \cong {\mathbb Z}_p\). Let \(\text Ш(E/{\mathbb Q}_n):=\ker \big(H^1({\mathbb Q}_n,E)\to \prod_v H^1({\mathbb Q}_{n,v},E)\big)\) be the Shafarevich-Tate group at level \(n\), where \(v\) varies over all places of \({\mathbb Q}_n\) and let \(p^{e_n}\) be its order. When \(p\) is ordinary, Mazur found [loc. cit.] that NEWLINE\[NEWLINE e_n-e_{n-1}=(p^n-p^{n-1})\mu +\lambda-r_{\infty} NEWLINE\]NEWLINE for \(n\gg 0\) and where \(r_{\infty}\) is the rank of \(E({\mathbb Q}_{\infty})\) which is an analogue of the result of Iwasawa for the class number of the \(n\)-th layer \({\mathbb Q}_n\) of \({\mathbb Q}_{\infty}/{\mathbb Q}\).NEWLINENEWLINEThe supersingular case, that is, when \(p\) divides \(a_p=p+1-\#E({ \mathbb F}_p)\) was studied by \textit{M. Kurihara} [Invent. Math. 149, No. 1, 195--224 (2002; Zbl 1033.11028)] and by \textit{B. Perrin-Riou} [Exp. Math. 12, No. 2, 155--186 (2003; Zbl 1061.11031)]. The case \(a_p=0\) has been more accessible. The general supersingular case has turned out to be more difficult. In this paper, the author provides three possible growth formulas involving algebraic Iwasawa invariants coming from two modified Selmer groups. He also generalizes Perrin-Riou's work giving the growth pattern of the Shafarevich-Tate group in the tower of number fields obtained by adjo…
reviewed by: Gabriel D. Villa Salvador
description: scientific article; zbMATH DE number 6211854