Ramification in the division fields of an elliptic curve
type: Item
instance of: scholarly article
title: Ramification in the division fields of an elliptic curve
zbMATH Open document ID: 1130.11323
DOI: 10.1007/BF02941276
author: Masanari Kida
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2004-05-18T00:00:00Z
review text: The author studies the ramification properties of the fields generated by torsion points of elliptic curves. In the first section, he starts with the following situation. Let \(p\) be a prime number, \(K\) a finite extension of the \(p\)-adic field \(\mathbb Q_p\) and \(\overline{K}\) an algebraic closure of \(K\). Let \(E\) be an elliptic curve defined over \(K\) and \(\ell\) a prime number distinct from \(p\). For any \(n\geq 1\), let us denote by \(E[\ell^n]\) the subgroup of the \(\ell^n\)-torsion points of \(E(\overline{K})\) and by \(K(E[\ell^n])\) the \(\ell^n\)-division field of \(E\) over \(K\). Let us recall that by the Néron-Ogg-Shafarevich criterion, \(E\) has good reduction over \(K\) if and only if for all \(n\geq 1\) the extension \(K(E[\ell^n])/K\) is unramified. Below the symbols \(I_\nu\), \(I_\nu^*\) \((\nu\geq 0)\) denote the Kodaira type of \(E\) over \(K\). The author proves the following results: 1) Suppose \(\ell\geq3\). The extension \(K(E[\ell])/K\) is unramified if and only if the Kodaira type of \(E\) is \(I_\nu\) with \(\nu\geq0\) and \(\nu\equiv0\bmod\ell\). 2) Suppose \(\ell=2\). The extension \(K(E[2])/K\) is unramified if and only if the Kodaira type of \(E\) is \(I_\nu\) or \(I_\nu^*\) with \(\nu\geq0\) and \(\nu\) even. 3) Assume that \(K(E[\ell])/K\) is unramified and \(E\) has bad reduction over \(K\). 3.1) If the Kodaira type of \(E\) is \(I_\nu\) \((\nu\geq1)\), then \(K(E[\ell^n])/K\) is unramified if and only if \(1\leq n\leq v_\ell(\nu)\) (\(v_\ell(\nu)\) being the \(\ell\)-adic valuation of \(\nu\)). 3.2) If the Kodaira type of \(E\) is \(I_\nu^*\) with \(\nu\geq0\), then \(K(E[\ell^n])/K\) is unramified if and only if \(n=1\). He also describes the pairs \((E,p)\) for which the extension \(K(E[\ell])/K\) is wildly ramified. In the second section, the author obtains results in case \(\ell=2\) and \(K=\mathbb Q_p\), \(p\) being any prime number. In the last section, as an application, he is interested in a characterization o…
reviewed by: Alain Kraus
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G07
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11S15
zbMATH DE Number: 2063943
zbMATH Keywords: elliptic curves
zbMATH Keywords: ramification
zbMATH Keywords: torsion points
describes a project that uses: ecdata
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2063943