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Transactions of the American Mathematical Society

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Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields


Authors: J.-L. Colliot-Thélène, D. Harbater, J. Hartmann, D. Krashen, R. Parimala and V. Suresh
Journal: Trans. Amer. Math. Soc. 372 (2019), 5263-5286
MSC (2010): Primary 14C25, 14G05, 14H25; Secondary 11E72, 12G05, 12F10
DOI: https://doi.org/10.1090/tran/7911
Published electronically: July 30, 2019
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Abstract: We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. We show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. Many of our results are shown to hold more generally in the henselian case.


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J.-L. Colliot-Thélène
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris–Sud, CNRS, Université Paris–Saclay, 91405 Orsay, France
Email: jlct@math.u-psud.fr

D. Harbater
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: harbater@math.upenn.edu

J. Hartmann
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: hartmann@math.upenn.edu

D. Krashen
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019
Email: daniel.krashen@rutgers.edu

R. Parimala
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: parimala@mathcs.emory.edu

V. Suresh
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: suresh@mathcs.emory.edu

DOI: https://doi.org/10.1090/tran/7911
Keywords: Linear algebraic groups and torsors, zero-cycles, local-global principles, semiglobal fields, discrete valuation rings
Received by editor(s): April 15, 2018
Published electronically: July 30, 2019
Additional Notes: The second and third authors were supported by NSF collaborative FRG grant DMS-1463733. The second author was also supported by NSF collaborative FRG grant DMS-1265290, and the third author by a Simons Fellowship.
The fourth author was supported by NSF collaborative FRG grant DMS-1463901. This author was also supported by NSF RTG grant DMS-1344994.
The fifth and sixth authors were supported by NSF collaborative FRG grant DMS-1463882. The fifth author was also supported by NSF grant DMS-1401319, and the sixth author by NSF grant DMS-1301785.
Article copyright: © Copyright 2019 American Mathematical Society