On cyclic twists of elliptic curves of period two or three and the determination of their relative Brauer groups
type: Item
instance of: scholarly article
title: On cyclic twists of elliptic curves of period two or three and the determination of their relative Brauer groups
zbMATH Open document ID: 1330.14056
author: Jung-Miao Kuo
published in: Journal of Pure and Applied Algebra
publication date: 2015-11-10T00:00:00Z
review text: For a smooth projective curve \(\mathcal{C}\) over a field \(k\), denote its function field by \(k(\mathcal{C})\). The \textit{relative Brauer group} of \(\mathcal{C}\), denoted by \(\text{Br}(k(\mathcal{C})/k)\), is the kernel of the natural homomorphism \(\text{Br}(k)\rightarrow \text{Br}(k(C))\). For a curve of genus zero, the structure of its relative Brauer group is well understood. If \(\mathcal{C}\) is a curve of genus one, then it is a principal homogenous space for some elliptic curve \(\mathcal{E}\) over \(k\), and it was shown by \textit{S. Lichtenbaum} [Am. J. Math. 90, 1209--1223 (1968; Zbl 0187.18602); Invent. Math. 7, 120--136 (1969; Zbl 0186.26402)] that if the period of \(\mathcal{C}\) equals its index, then the elements of \(\text{Br}(k(\mathcal{C})/k)\) can be parameterized by the group \(\mathcal{E}(k)\) of \(k\)-rational points of \(\mathcal{E}\). However, Lichtenbaum's mapping is difficult to apply for computation. In a recent work, \textit{M. Ciperiani} and \textit{D. Krashen} [Isr. J. Math. 192, Part B, 921--949 (2012; Zbl 1259.14020)] described the parametrization of \(\text{Br}(k(\mathcal{C})/k)\) by \(\mathcal{E}(k)\) in terms of a cup product formula when \(\mathcal{C}\) is what they call a \textit{cyclic twist} of \(\mathcal{E}\). For the convenience of the reader, we quote the definition used by the author of the paper under review: ``A principal homogeneous space \(\mathcal{C}\) for an elliptic curve \(\mathcal{E}\) over \(k\) is a cyclic twist if there is a finite cyclic subgroup \(\mathcal{T}\) of the group \(\mathcal{E}\) of order \(n=\text{per}(\mathcal{C})\) stable under the action of the absolute Galois group \(G_k=\text{Gal}(k_s/k)\), where \(k_s\) is a fixed separable closure of \(k\), such that the cohomology class in \(H^1(k,\mathcal{E})\) corresponding to \(\mathcal{C}\) can be represented as the image of a class in \(H^1(k,\mathcal{T})\) under the map induced by the inclusion \(\mathcal{T}\hookrightarrow \mathcal{E}\).'' It…
reviewed by: Cherng-tiao Perng
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 6506982
zbMATH Keywords: elliptic curve
zbMATH Keywords: relative Brauer group
zbMATH Keywords: cyclic twist
describes a project that uses: RelBrauer
MaRDI profile type: Publication
full work available at URL: j.jpaa.2015.08.016
OpenAlex ID: W2176117760
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DOI: 10.1016/J.JPAA.2015.08.016
description: scientific article; zbMATH DE number 6506982