5-torsion in the Shafarevich-Tate group of a family of elliptic curves
type: Item
instance of: scholarly article
title: 5-torsion in the Shafarevich-Tate group of a family of elliptic curves
zbMATH Open document ID: 0983.11029
author: Cheryl DeLorme Beaver
published in: Journal of Number Theory
publication date: 2002-04-10T00:00:00Z
review text: Let \(E\) be an elliptic curve over a number field \(k\). There is an abelian group attached to \(E\) known as the Shafarevich-Tate group of \(E/k\), and it is equipped with an alternating pairing known as the Cassels-Tate pairing. This pairing is important because if the Shafarevich-Tate group is finite (as it is conjectured to be), the pairing lets one decide whether principal homogeneous spaces of \(E\) are trivial or not, and the latter is needed to compute Mordell-Weil ranks via descent. Even without assuming the finiteness of the Shafarevich-Tate group, the pairing can be used to construct nontrivial elements of that group. The paper under review considers the following special case. Let \(E'\) be an elliptic curve over \(\mathbb Q\) whose \(5\)-torsion subgroup is isomorphic to \({\mathbb Z}/5 \oplus \mu_5\) as a Galois module. Let \(E\) be the quotient of \(E'\) by \({\mathbb Z}/5\), so that there is an isogeny \(\phi: E \rightarrow E'\) with kernel isomorphic to \(\mu_5\). The restriction of the Cassels-Tate pairing for \(E\) to the \(\phi\)-torsion in the Shafarevich-Tate group can be viewed as a pairing on the \(\phi\)-Selmer group \(S_\phi(E/{\mathbb Q})\). Descent theory implies that \(S_\phi(E/{\mathbb Q})\) can be identified with a finite subgroup of \({\mathbb Q}^*/{\mathbb Q}^{*5}\), and that this subgroup is effectively computable, at least in theory. The author first gives a very concrete description of the group \(S_\phi(E/{\mathbb Q})\) as a subgroup of \({\mathbb Q}^*/{\mathbb Q}^{*5}\) (Theorem 1.1), but her main result is that there is also a practical formula for the pairing on it in terms of local Hilbert norm residue symbols (Theorem 1.2). The precise statements and proofs of the theorems are expressed using the universal family of elliptic curves \(E'\) with \(5\)-torsion above, which was computed by \textit{K. Rubin} and \textit{A. Silverberg} [in Elliptic curves, modular forms, and Fermat's Last Theorem, 148-161 (1995; Zbl 0856.11027)].…
reviewed by: Bjorn Poonen
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 1538252
zbMATH Keywords: Shafarevich-Tate group
zbMATH Keywords: Cassels-Tate pairing
zbMATH Keywords: elliptic curve
zbMATH Keywords: 5-torsion
zbMATH Keywords: Selmer group
MaRDI profile type: Publication
full work available at URL: jnth.1999.2493
OpenAlex ID: W2049677520
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DOI: 10.1006/JNTH.1999.2493
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description: scientific article; zbMATH DE number 1538252