On Néron's classification of elliptic curves of residue characteristics 2 and 3
type: Item
instance of: scholarly article
title: On Néron's classification of elliptic curves of residue characteristics 2 and 3
zbMATH Open document ID: 0786.14020
author: Juan-Miguel Gracia
published in: Journal of Number Theory
publication date: 1993-10-18T00:00:00Z
review text: In \textit{A. Néron}'s famous paper ``Modèles minimaux des variétés abéliennes sur les corps locaux et globaux'' in Publ. Math., Inst. Hautes Étud. Sci. 21 (1964; Zbl 0132.414) a classification of reduction types of elliptic curves \(E\) over a local field \(K\) with normalized additive valuation \(v\) is given. Tate invented an algorithm for determining the reduction type of a minimal model of \(E\) over \(K\) by applying some suitable birational transformations to a Weierstrass model of \(E\) [\textit{J. Tate} in Modular functions of one variable, IV. Proc. internat. summer school, Univ. Antwerp, RUCA 1972, Lect. Notes Math. 476 (1975; Zbl 0315.14014)]. The present paper is aimed at characterizing the various reduction types of \(E\) over \(K\) solely in terms of the discriminant \(\Delta\) of \(E\) and of the derived coefficients \(c_ 4\), \(c_ 6\), obtained by Tate's formulas from the coefficients \(a_ i\) of the Weierstrass equation \[ y^ 2+a_ 1xy+a_ 3y=x^ 3+a_ 2x^ 2+a_ 4x+a_ 6\quad (a_ i\in K) \] of \(E\). The field \(K\) is assumed to be of characteristic zero. In case the residue field \(k\) of \(K\) with respect to the valuation \(v\) is of characteristic \(\neq 2,3\), it is known that the values \(v(c_ 4)\), \(v(c_ 6)\) and \(v(\Delta)\) suffice to characterize the reduction type (see table I). In the remaining cases of characteristic 2 or 3, however, this is not always true, and the author therefore finds some extra (congruence) conditions which together with the \(v\)-values enable him to distinguish the various possible reduction types (see tables II--V). Such a characterization of the reduction types in terms of the coefficients is an indispensable tool in working with elliptic curves \(E\) over local and global fields. Beside Tate's algorithm, a corresponding algorithm of \textit{M. Laska} [Math. Comput. 38, 257-260 (1982; Zbl 0493.14016)] should be mentioned in this connection, too.
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 11G07
zbMATH DE Number: 431315
zbMATH Keywords: elliptic curves over a local field
zbMATH Keywords: algorithm for the reduction type of a minimal model
zbMATH Keywords: characteristic 2
zbMATH Keywords: characteristic 3
zbMATH Keywords: Weierstrass model
zbMATH Keywords: 315
reviewed by: Horst G. Zimmer
MaRDI profile type: Publication
full work available at URL: jnth.1993.1040
OpenAlex ID: W2006356656
DOI: 10.1006/JNTH.1993.1040
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description: scientific article; zbMATH DE number 431315