Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-11T06:03:30.120Z Has data issue: false hasContentIssue false
  • English
  • Français

HASSE PRINCIPLES FOR ÉTALE MOTIVIC COHOMOLOGY

Part of: Arithmetic algebraic geometry (Co)homology theory

Published online by Cambridge University Press:  26 December 2018

THOMAS H. GEISSER Open the ORCID record for THOMAS H. GEISSER [Opens in a new window]
THOMAS H. GEISSER*
Affiliation:
Rikkyo University, Ikebukuro, Tokyo, Japan email geisser@rikkyo.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss the kernel of the localization map from étale motivic cohomology of a variety over a number field to étale motivic cohomology of the base change to its completions. This generalizes the Hasse principle for the Brauer group, and is related to Tate–Shafarevich groups of abelian varieties.

MSC classification

Primary: 11G35: Varieties over global fields 14F22: Brauer groups of schemes 14F42: Motivic cohomology; motivic homotopy theory
Type
Article
Information
Nagoya Mathematical Journal , Volume 236: Celebrating the 60th Birthday of Shuji Saito , December 2019 , pp. 63 - 83
Copyright
© 2018 Foundation Nagoya Mathematical Journal  

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported by JSPS Grant-in-Aid (C) 18K03258, (A) 15H02048-1.

References

Bloch, S., The moving lemma for higher Chow groups , J. Algebraic Geom. 3(3) (1994), 537–568.Google Scholar
Feng, T., Etale Steenrod operations and the Artin-Tate pairing, preprint, 2017, arXiv:1706.00151.Google Scholar
Gabber, O., Sur la torsion dans la cohomologie l-adique d’une variétè , C. R. Math. Acad. Sci. Paris Sér. I Math. 297(3) (1983), 179–182.Google Scholar
Geisser, T., Motivic cohomology over Dedekind rings , Math. Z. 248(4) (2004), 773–794.Google Scholar
Geisser, T., Weil-etale cohomology , Math. Ann. 330 (2004), 665–692.10.1007/s00208-004-0564-8Google Scholar
Geisser, T., On the structure of étale motivic cohomology , J. Pure Appl. Algebra 221(7) (2017), 1614–1628.10.1016/j.jpaa.2016.12.019Google Scholar
Geisser, T., Comparing the Brauer group and the Tate-Shafarevich group, J. Inst. Math. Jussieu, to appear, arXiv:1712.06249.Google Scholar
Geisser, T., Tate’s conjecture and the Tate-Shafarevich group over global function fields, preprint, 2018, arXiv:1801.02406.10.1017/S147474801900046XGoogle Scholar
Geisser, T. and Levine, M., The K-theory of fields in characteristic p , Invent. Math. 139(3) (2000), 459–493.10.1007/s002220050014Google Scholar
Geisser, T. and Levine, M., The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky , J. Reine Angew. Math. 530 (2001), 55–103.Google Scholar
Geisser, T. and Schmidt, A., Poitou-Tate duality for arithmetic schemes , Compos. Math. 154(9) (2018), 2020–2044.Google Scholar
González-Avilés, C. D. and Tan, K.-S., A generalization of the Cassels-Tate dual exact sequence , Math. Res. Lett. 14(2) (2007), 295–302.10.4310/MRL.2007.v14.n2.a11Google Scholar
Grothendieck, A., “ Le groupe de Brauer. III. Exemples et compléments ”, in Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3 , North-Holland, Amsterdam, 1968, 88–188.Google Scholar
Jannsen, U., Continuous étale cohomology , Math. Ann. 280(2) (1988), 207–245.10.1007/BF01456052Google Scholar
Jannsen, U., “ On the l-adic cohomology of varieties over number fields and its Galois cohomology ”, in Galois Groups Over ℚ (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 16 , Springer, New York, 1989, 315–360.10.1007/978-1-4613-9649-9_5Google Scholar
Katz, N. and Lang, S., Finiteness theorems in geometric classfield theory , Enseign. Math. (2) 27(3–4) (1981), 285–319; With an appendix by Kenneth A. Ribet. (1982).Google Scholar
Kriz, I., On the arithmetic of elliptic curves and a homotopy limit problem , J. Number Theory 183 (2018), 466–484.10.1016/j.jnt.2017.08.008Google Scholar
Lichtenbaum, S., “ Values of zeta-functions at nonnegative integers ”, in Number Theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Mathematics 1068 , Springer, Berlin, 1984, 127–138.Google Scholar
Liu, Q., Lorenzini, D. and Raynaud, M., Néron models, Lie algebras, and reduction of curves of genus one , Invent. Math. 157(3) (2004), 455–518.Google Scholar
Milne, J.S., Values of zeta functions of varieties over finite fields , Amer. J. Math. 108(2) (1986), 297–360.10.2307/2374676Google Scholar
Milne, J.S., Arithmetic duality theorems. Second edition. BookSurge, LLC, Charleston, SC, 2006. viii $+$ 339 pp. ISBN: 1-4196-4274-X.Google Scholar
Milne, J.S., Etale Cohomology, Princeton Mathematical Series 33 , Princeton University Press, Princeton, NJ, 1980.10.1515/9781400883981Google Scholar
Poonen, B., Rational Points on Varieties, Graduate Studies in Mathematics 186 , American Mathematical Society, Providence, RI, 2017.10.1090/gsm/186Google Scholar
Poonen, B. and Stoll, M., The Cassels-Tate pairing on polarized abelian varieties , Ann. of Math. (2) 150(3) (1999), 1109–1149.Google Scholar
Saito, S., “ A global duality theorem for varieties over global fields ”, in Algebraic K-theory: Connections with Geometry and Topology (Lake Louise, AB, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279 , Kluwer, Dordrecht, 1989, 425–444.10.1007/978-94-009-2399-7_14Google Scholar
Saito, S., Arithmetic on two-dimensional local rings , Invent. Math. 85(2) (1986), 379–414.Google Scholar
Schneider, P., Über gewisse Galoiscohomologiegruppen , Math. Z. 168(2) (1979), 181–205.10.1007/BF01214195Google Scholar
Voevodsky, V., On motivic cohomology with ℤ/l-coefficients , Ann. of Math. (2) 174 401–438.Google Scholar