Torsion points on elliptic curves and Galois module structure
type: Item
instance of: scholarly article
title: Torsion points on elliptic curves and Galois module structure
zbMATH Open document ID: 0864.11055
DOI: 10.1007/BF01232369
published in: Inventiones Mathematicae
publication date: 1997-06-22T00:00:00Z
full work available at URL: https://eudml.org/doc/144338
review text: Let \(E\) be an elliptic curve defined over a number field \(F\) with everywhere good reduction. For a prime \(p>3\) let \({\mathfrak B}_i(F)\) denote the \({\mathcal O}_F\)-Hopf algebra which belongs to the \({\mathcal O}_F\)-group scheme of \(p^i\)-torsion points on \(E\). Let furthermore \({\mathfrak A}_i(F)\) be its Cartier dual. For any point \(Q\in E(F)\) let \(G_Q(i)= \{Q'\in E (\overline F) \mid p^iQ'=Q\}\) and the Kummer algebra \(F_Q(i)= \text{Map} (G_Q(i), \overline F)^{\text{Gal} (\overline F/F)}\). One can now consider the \({\mathfrak A}_i(F)\)-module \({\mathfrak C}_Q(i)\) which is the ``modified'' integral closure of \({\mathcal O}_F\) in \(F_Q(i)\). The author proves that \({\mathfrak C}_Q(i)\) is a globally free \({\mathfrak A}_i(F)\)-module if \(Q\) is a torsion point in \(E(F)\). This generalizes a result of \textit{A. Srivastav} and \textit{M. J. Taylor} [Invent. Math. 99, 165-184 (1990; Zbl 0705.14031)] who proved the same result under the additional assumption that \(E\) has complex multiplication. The proof depends upon the use of intersection theory on arithmetic surfaces.
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 871765
zbMATH Keywords: \(p^ i\)-torsion points
zbMATH Keywords: modified integral closure
zbMATH Keywords: Galois module structure
zbMATH Keywords: elliptic curve
author: A. Agboola
MaRDI profile type: Publication
cites work: Q4012330
cites work: Q3758001
cites work: Q3994900
cites work: Q3710637
cites work: Elliptic curves with complex multiplication and Galois module structure
cites work: Mordell-Weil groups and the Galois module structure of rings of integers
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description: scientific article; zbMATH DE number 871765