Curves that are cyclic twists of and the relative Brauer groups

Authors:
Darrell E. Haile, Ilseop Han and Adrian R. Wadsworth

Journal:
Trans. Amer. Math. Soc. **364** (2012), 4875-4908

MSC (2010):
Primary 16K50

DOI:
https://doi.org/10.1090/S0002-9947-2012-05526-7

Published electronically:
April 25, 2012

MathSciNet review:
2922613

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a field with . Let be the projective curve of a binary cubic form , and the function field of . In this paper we explicitly describe the relative Brauer group of over . When is diagonalizable we show that every algebra in is a cyclic algebra obtainable using the -coordinate of a -rational point on the Jacobian of . But when is not diagonalizable, the algebras in 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 , the rationals, and for , where is a primitive third root of unity. The approach is to realize as a cyclic twist of its Jacobian , 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:
https://doi.org/10.1090/S0002-9947-2012-05526-7

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.