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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Algebraic families of nonzero elements
of Shafarevich-Tate groups


Authors: Jean-Louis Colliot-Thélène and Bjorn Poonen
Journal: J. Amer. Math. Soc. 13 (2000), 83-99
MSC (1991): Primary 11G10; Secondary 11G30, 11G35, 14H40, 14J27
Published electronically: August 20, 1999
MathSciNet review: 1697093
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Abstract | References | Similar Articles | Additional Information

Abstract: Principal homogeneous spaces under an abelian variety defined over a number field $k$ may have rational points in all completions of the number field without having rational points over $k$. Such principal homogeneous spaces represent the nonzero elements of the Shafarevich-Tate group of the abelian variety.

We produce nontrivial, one-parameter families of such principal homogeneous spaces. The total space of these families are counterexamples to the Hasse principle. We show these may be accounted for by the Brauer-Manin obstruction.


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Additional Information

Jean-Louis Colliot-Thélène
Affiliation: C.N.R.S., Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay, France
Email: colliot@math.u-psud.fr

Bjorn Poonen
Affiliation: C.N.R.S., Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay, France
Email: poonen@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-99-00315-X
PII: S 0894-0347(99)00315-X
Keywords: Shafarevich-Tate group, Brauer-Manin obstruction, Hasse principle, cubic surface, Cassels-Tate pairing, Lefschetz pencil
Received by editor(s): January 8, 1999
Received by editor(s) in revised form: June 9, 1999
Published electronically: August 20, 1999
Additional Notes: Most of the research for this paper was done while the authors were both enjoying the hospitality of the Isaac Newton Institute, Cambridge, England. The first author is a researcher at C.N.R.S. The second author is partially supported by NSF grant DMS-9801104, a Sloan Fellowship, and a Packard Fellowship.
Article copyright: © Copyright 1999 American Mathematical Society