The index of certain hyperelliptic curves over \(p\)-adic fields
type: Item
instance of: scholarly article
title: The index of certain hyperelliptic curves over \(p\)-adic fields
zbMATH Open document ID: 1039.11043
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2004-01-22T00:00:00Z
review text: Let \(K\) be a finite extension of the field of \(p\)-adic numbers \(\mathbb{Q}_p\) and \(O_K\) be the ring of integers in \(K\). Let \(C\) be a geometrically connected smooth projective curve over \(K\). The index \(I(C)\) of \(C\) is defined to be the greatest common divisor of the divisors on \(C\). In this paper, which is the sequel of [\textit{J. Van Geel} and \textit{V. I. Yanchevskii}, Manuscr. Math. 96, 317--333 (1998; Zbl 1015.11023) and Ann. Fac. Sci. Toulouse, VI. Sér., Math. 8, 155--172 (1999; Zbl 0986.11036)], the authors deal with the index of curves \(C\) defined by an affine equation of the form \(Y^2= \pi f(X)\), where \(\pi\) is a uniformizing element and \(f(X)\) a monic irreducible polynomial of \(O_K[X]\). They give necessary and sufficient conditions for \(I(C)\) to be 1 and 2 in cases where \(K\) is dyadic and non-dyadic, respectively.
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 2031028
zbMATH Keywords: hyperelliptic curves
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: index of a curve
author: Jan Van Geel
author: V. I. Yanchevskiĭ
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2031028