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), 48754908
MSC (2010):
Primary 16K50
Published electronically:
April 25, 2012
MathSciNet review:
2922613
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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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 L. E. Dickson, Algebraic Invariants, J. Wiley, New York, 1914.
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 B. H. Gross, Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Mathematics, 776, Springer, Berlin, 1980. MR 563921 (81f:10041)
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 D. Haile, On Clifford algebras, conjugate splittings, and function fields of curves, in Israel Math.Conf.Proc, volume 1: Ring Theory 1989, in honor of S.A. Amitsur (L. Rowen, ed.) Weizman Science Press, Jerusalem, 356361. MR 1029325 (91c:11065)
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 D. Haile, When is the Clifford algebra of a binary cubic form split?, J.Algebra, 146 (1992), 514520. MR 1152918 (93a:11029)
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 [HKRT]
 D. Haile, M.A. Knus, M. Rost, and J.P. Tignol, Algebras of odd degree with involution, trace forms and dihedral extensions, Israel J. Math., 96 (1996), 299340. MR 1433693 (98h:16024)
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 [KMRT]
 M.A. Knus, A. S. Merkurjev, M. Rost, and J.P. Tignol, The Book of Involutions, Amer.Math.Soc., Providence, RI, 1998. MR 1632779 (2000a:16031)
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 S. Lichtenbaum, Duality theorems for curves over adic fields, Invent. Math., 7 (1969), 120136. MR 0242831 (39:4158)
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 J. S. Milne, Class Field Theory, http://www.jmilne.org/math/CourseNotes/CFT400.pdf.
 [SAGE]
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 J.P. Serre, Local Fields, Springer, New York, 1979. MR 554237 (82e:12016)
 [Si]
 J. Silverman, The Arithmetic of Elliptic Curves, Springer, New York, 1986. MR 817210 (87g:11070)
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 J. Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris, Sér. AB, 273 (1971), A238A241.
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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 920930112
Email:
arwadsworth@ucsd.edu
DOI:
http://dx.doi.org/10.1090/S000299472012055267
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.
