Elliptic curves with large Tate-Shafarevich groups over a number field
type: Item
instance of: scholarly article
title: Elliptic curves with large Tate-Shafarevich groups over a number field
zbMATH Open document ID: 1191.11018
author: Kazuo Matsuno
published in: Mathematical Research Letters
publication date: 2009-09-29T00:00:00Z
review text: In this nice paper the author studies the unboundedness of the Tate-Shafarevich group in a family of elliptic curves defined over a fixed number field. The problem of determining how big this group can be for various elliptic curves is classical and has extensively been studied by Cassels, Kloosterman, Kramer and many others (see for instance \textit{J. W. S. Cassels} [J. Reine Angew. Math. 214--215, 65--70 (1964; Zbl 0236.14012)], \textit{R. Kloosterman} [J. Théor. Nombres Bordx. 17, No. 3, 787--800 (2005; Zbl 1153.11313)], \textit{K. Kramer} [Proc. Am. Math. Soc. 89, 379--386 (1983; Zbl 0567.14018)] and references there). A refinement of the above question is to study, for a fixed prime \(p\), the unboundedness of the \(p\)-part of the Tate-Shafarevich group. In this direction, for \(p=3\), Cassels gave a positive answer in the paper cited above showing that the \(3\)-part of \(\text Ш(E/\mathbb{Q})\) is unbounded if \(E\) runs through all elliptic curves with \(j\)-invariant \(0\). This result was eventually generalized in [\textit{N. Aoki}, Acta Arith. 112, No. 3, 209--227 (2004; Zbl 1083.11035); \textit{R. Bölling}, Math. Nachr. 67, 157--179 (1975; Zbl 0314.14008); \textit{T. Fisher}, J. Eur. Math. Soc. (JEMS) 3, No. 2, 169--201 (2001; Zbl 1007.11031); \textit{R. Kloosterman} and \textit{E. F. Schaefer}, J. Number Theory 99, No. 1, 148--163 (2003; Zbl 1074.11032); \textit{K. Matsuno}, Manuscr. Math. 122, No. 3, 289--304 (2007; Zbl 1152.11045)] to \(p\leq 7\) and \(p=13\). The main technique employed there was to produce infinitely many elliptic curves over \(\mathbb{Q}\) with a \(p\)-isogeny and to study the effect of these isogenies on the Tate-Shafarevich group. Since the genus of \(X_0(p)\) is greater than \(1\) for \(p\geq 23\), for such primes there are only finitely many elliptic curves over \(\mathbb{Q}\) with a \(p\)-isogeny defined over \(\mathbb{Q}\); therefore the above strategy breaks down for large primes. Observe that this obstruction occurs also …
reviewed by: Filippo Nuccio
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 5608655
MaRDI profile type: Publication
full work available at URL: mrl.2009.v16.n3.a6
OpenAlex ID: W2061178623
DOI: 10.4310/MRL.2009.V16.N3.A6
description: scientific article; zbMATH DE number 5608655