On cyclic twists of elliptic curves of period two or three and the determination of their relative Brauer groups
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 was shown by Ciperiani and Krashen [loc. cit.] that if \(\mathcal{C}\) is a cyclic twist of \(\mathcal{E}\), then \(\text{per}(\mathcal{C})=\text{in}(\mathcal{C})\) (i.e period and index of \(\mathcal{C}\) agree). As the title of the paper under review shows, the author studies cyclic twists of elliptic curves of period two or three, and determines their relative Brauer groups. Section 1 of the paper is an introduction. In Section 2, the author recalls the key results of Ciperiani and Krashen that will be needed, and also a result of \textit{J. Vélu} [C. R. Acad. Sci., Paris, Sér. A 273, 238--241 (1971; Zbl 0225.14014)] that gave explicit descriptions for the isogeny in question, and the Weierstrass equations for the corresponding elliptic curves. Section 3 studies cyclic twists of elliptic curves of period three. The main result of it is as follows. { Theorem 3.6.} Let \(f\) be a nondegenerate binary cubic from over \(k\) and let \(\mathcal{C}_f\) be the curve over \(k\) given by the equation \[ \mathcal{C}_f:Z^3-\lambda h_f(X,Y)Z-f(X,Y)=0, \] where \(\lambda\in k\) and \(h_f\) is the Hessian determinant of \(f\). Then \(\mathcal{C}_f\) is a cyclic twist of its Jacobian. Then Section 4 and Section 5 give detailed and explicit descriptions of the relative Brauer groups of cyclic twists of period three, and period two (some case of genus one hyperelliptic curves). Both of these sections rely on the results of Ciperiani and Krashen, and those of Vélu mentioned in Section 2. The author illustrates the results with concrete examples. It is noted in the end of Section 5 that the relative Brauer group of arbitrary genus one hyperelliptic curve has been determined explicitly by \textit{D. Haile} and \textit{I. Han} [J. Algebra 313, No. 2, 811--823 (2007; Zbl 1121.14022)] with different arguments which are briefly sketched.
scientific article; zbMATH DE number 6506982
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