On quadratic twists of hyperelliptic curves
type: Item
instance of: scholarly article
title: On quadratic twists of hyperelliptic curves
zbMATH Open document ID: 1304.14041
DOI: 10.1216/RMJ-2014-44-3-1015
author: Mohammad Sadek
published in: Rocky Mountain Journal of Mathematics
publication date: 2014-09-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.0732
full work available at URL: https://projecteuclid.org/euclid.rmjm/1411945677
reviewed by: Fernando Torres
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 6349623
zbMATH Keywords: hyperelliptic curves
zbMATH Keywords: quadratic twist
MaRDI profile type: Publication
arXiv ID: 1010.0732
cites work: Stable reduction of curves and tame ramification
cites work: Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète
cites work: Q4530238
cites work: Models of curves and wild ramification
cites work: Wild quotient singularities of surfaces
cites work: Dual graphs of degenerating curves
cites work: Problems in Algebraic Number Theory
cites work: The complete classification of fibres in pencils of curves of genus two
cites work: MINIMAL MODELS OF CURVES OF GENUS 2 AND HOMOMORPHISMS OF ABELIAN VARIETIES DEFINED OVER A FIELD OF FINITE CHARACTERISTIC
cites work: Minimal resolution of Atkin-Lehner quotients of \(X_0(N)\)
Recommended article: Modèles minimaux des courbes de genre deux.
Recommended article: N-Covers of hyperelliptic curves
Recommended article: Arithmetic of hyperelliptic curves over local fields
Recommended article: On periodicity of continued fractions in hyperelliptic function fields
Recommended article: Minimal regular models of quadratic twists of genus two curves
Recommended article: Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group
Recommended article: How to distinguish hyperelliptic curves in even characteristic.
Recommended article: Hyperelliptic curves of genus three over finite fields of even characteristic
Recommended article: A family of maximal hyperelliptic curves
review text: Let \(K\) be a field with \(\text{char}(K)\neq 2\), \(f(x)\in K[x]\) a separable polynomial and \(C|K:\, y^2=f(x)\) a hyperelliptic curve of genus \(g\geq 2\) over \(K\) (thus \(\text{deg}(f)\in\{2g+1,2g+2\}\)). Let \(d\in K\setminus K^2\) and set \(K':=K(\sqrt{d})\), which becomes a separable quadratic extension of \(K\). Let \(C_d\) be the hyperelliptic quadratic twist obtained from \(C|K'\) and which is defined by an equation of type \(dy^2=f(x)\); in particular, \(C\) and \(C_d\) are isomorphic over \(K'\). In this context a good reference is [\textit{Q. Liu}, Algebraic geometry and arithmetic curves. Transl. by Reinie Erné. Oxford: Oxford University Press (2006; Zbl 1103.14001)].NEWLINENEWLINEFor \(K\) a complete discrete valuation field, the main result of this paper is concerning minimal regular models of \(C_d\) which mainly depends on the fact that \(C(K)\neq\emptyset\). Then a nice application is giving, namely on the existence of hyperelliptic curves \(C\) of positive genus defined over \(\mathbb Q\), the existence of certain integers \(d\) such that \(C_d(\mathbb Q_p)=C_d(\mathbb Q)=\emptyset\) for infinitely many prime numbers \(p\).
description: scientific article; zbMATH DE number 6349623