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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Curves $ \mathcal{C}$ that are cyclic twists of $ Y^{2}=X^{3}+c$ and the relative Brauer groups $ Br(k(\mathcal{C})/k)$


Authors: Darrell E. Haile, Ilseop Han and Adrian R. Wadsworth
Journal: Trans. Amer. Math. Soc. 364 (2012), 4875-4908
MSC (2010): Primary 16K50
Published electronically: April 25, 2012
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Abstract: Let $ k$ be a field with $ {\sl {char}}(k) \ne 2,3$. Let $ \mathcal {C}_{f}$ be the projective curve of a binary cubic form $ f$, and $ k(\mathcal {C}_{f})$ the function field of $ \mathcal {C}_{f}$. In this paper we explicitly describe the relative Brauer group $ {\sl {Br}}(k(\mathcal {C}_{f})/k)$ of $ k(\mathcal {C}_{f})$ over $ k$. When $ f$ is diagonalizable we show that every algebra in $ {\sl {Br}}(k(\mathcal {C}_{f})/k)$ is a cyclic algebra obtainable using the $ y$-coordinate of a $ k$-rational point on the Jacobian $ \mathcal {E}$ of $ \mathcal {C}_{f}$. But when $ f$ is not diagonalizable, the algebras in $ {\sl {Br}}(k(\mathcal {C}_{f})/k)$ are presented as cup products of cohomology classes, but not as cyclic algebras. In particular, we provide several specific examples of relative Brauer groups for $ k=\mathbb{Q}$, the rationals, and for $ k=\mathbb{Q}(\omega )$, where $ \omega $ is a primitive third root of unity. The approach is to realize $ \mathcal {C}_{f}$ as a cyclic twist of its Jacobian $ \mathcal {E}$, an elliptic curve, and then apply a recent theorem of Ciperiani and Krashen.


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Additional Information

Darrell E. Haile
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: haile@indiana.edu

Ilseop Han
Affiliation: Department of Mathematics, California State University, San Bernardino, California 92407
Email: ihan@csusb.edu

Adrian R. Wadsworth
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112
Email: arwadsworth@ucsd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05526-7
PII: S 0002-9947(2012)05526-7
Received by editor(s): April 5, 2010
Received by editor(s) in revised form: December 1, 2010
Published electronically: April 25, 2012
Additional Notes: We would like to thank D. Krashen for some useful conversations, and in particular for his assistance with the proof of Theorem 6.1.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.