Indices of double coverings of genus 1 over \(p\)-adic fields
type: Item
instance of: scholarly article
title: Indices of double coverings of genus 1 over \(p\)-adic fields
zbMATH Open document ID: 0986.11036
author: Jan Van Geel
author: V. I. Yanchevskiĭ
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2002-06-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73476
review text: Let \(K\) be a local non-dyadic field of characteristic zero and \(O_K\) the valuation ring in \(K\). Consider the curve \(C_h\) defined by the equation \(Y^2=h(X)\), where \(h(X)\) is a polynomial of \(K[X]\) with no multiple roots. The index of \(C_h\), \(I(C_h)\), is by definition the greatest common divisor of the degrees of all \(K\)-rational divisors on \(C_h\). Let \(\pi\) be a uniformising element of \(K\) and \(\alpha\) a unit of \(O_K\) which is not a square in \(K\). In [Manuscr. Math. 96, 317--333 (1998; Zbl 1015.11023)] the authors determined the index of the elliptic curves \(C_{\varepsilon f}\), where \(\varepsilon\in \{1,\alpha,\pi, \alpha\pi\}\) and \(f(X)\) is a monic polynomial of \(O_K[X]\) which is not of the form \(f(X)= f_1(X) f_2(X)\), where \(f_1(X), f_2(X)\) are irreducible monic polynomials of \(O_K[X]\). In this paper the authors determine the index of the elliptic curves \(C_{\varepsilon f}\) for such polynomials \(f(X)\).
Mathematics Subject Classification ID: 11G07
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 1491219
zbMATH Keywords: double coverings of genus 1
zbMATH Keywords: local non-dyadic field
zbMATH Keywords: monic polynomial
zbMATH Keywords: index
zbMATH Keywords: elliptic curves
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
OpenAlex ID: W2329273979
cites work: Q5521595
cites work: Duality theorems for curves over p-adic fields
cites work: Q3708857
cites work: Q5509718
cites work: Indices of hyperelliptic curves over \(p\)-adic fields
cites work: Q4886199
DOI: 10.5802/AFST.927
description: scientific article; zbMATH DE number 1491219