ABC and the Hasse principle for quadratic twists of hyperelliptic curves
type: Item
instance of: scholarly article
title: ABC and the Hasse principle for quadratic twists of hyperelliptic curves
zbMATH Open document ID: 1404.11091
author: Pete L. Clark
author: Lori D. Watson
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2018-09-06T00:00:00Z
review text: Let \(C\) be an irreducible smooth curve, defined over \({\mathbb Q}\), admitting an involution \(\iota:C \to C\) (that is, an automorphism of order two). For each \(d \in {\mathbb Q}^{\times}/{\mathbb Q}^{\times 2}\) there is an irreducible smooth curve \({\mathcal T}_{d}(C, \iota)\), also defined over \({\mathbb Q}\), which is isomorphic to \(C\) (the isomorphism defined over \({\mathbb Q}(\sqrt{d})\)). It is said that \({\mathcal T}_{d}(C, \iota)\) violates the Hasse principle if \({\mathcal T}_{d}(C, \iota)({\mathbb A}_{\mathbb Q}) \neq \emptyset\) and \({\mathcal T}_{d}(C, \iota)({\mathbb Q})=\emptyset\), where \({\mathbb A}_{\mathbb Q}\) is the adele ring over \({\mathbb Q}\). In the paper [\textit{P. L. Clark} and \textit{J. Stankewicz}, Proc. Am. Math. Soc. 146, No. 7, 2839--2851 (2018; Zbl 1448.11113)] there is provided a criteria on \(C\) and \(\iota\) in order to have infinitely many values of \(d \in {\mathbb Q}^{\times}/{\mathbb Q}^{\times 2}\) such that \({\mathcal T}_{d}(C, \iota)\) violates the Hasse principle. That criteria is given by four conditions. One of these conditions is to assume \((C/\iota)({\mathbb Q})\) to be finite. If \(C\) is hyperelliptic and \(\iota\) is the hyperelliptic involution, one has that \((C/\iota)({\mathbb Q})={\mathbb P}^{1}({\mathbb Q})\) is infinite, so one cannot use the above criteria in this case. In the paper under review, the authors prove, under the assumption of the ABC conjecture, that if \(C\) is a hyperelliptic curve of genus at least three, then (i) the hyperelliptic involution \(\iota\) has no \({\mathbb Q}\)-rational points if and only if (ii) there are infinitely many values of \(d \in {\mathbb Q}^{\times }/{\mathbb Q}^{\times 2}\) such that \({\mathcal T}_{d}(C, \iota)\) violates the Hasse principle if and only if (iii) there some \(d \in {\mathbb Q}^{\times }/{\mathbb Q}^{\times 2}\) such that \({\mathcal T}_{d}(C, \iota)\) violates the Hasse principle. The hard part is to prove that (i) implies (ii). I…
reviewed by: Rubén A. Hidalgo
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 6933366
zbMATH Keywords: ABC conjecture
zbMATH Keywords: quadratic twists
zbMATH Keywords: Hasse principle
MaRDI profile type: Publication
full work available at URL: j.crma.2018.07.007
OpenAlex ID: W2886812330
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Wikidata QID: Q129421785
DOI: 10.1016/J.CRMA.2018.07.007
description: scientific article; zbMATH DE number 6933366