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On the ℓ-adic cohomology of varieties over number fields and its Galois cohomology

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Galois Groups over ℚ

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 16))

Abstract

If X is a smooth, projective variety over a number field k, then the absolute Galois group Gk = Gal(k̄/k) acts on the étale cohomology groups Hi(X̄, ℚℓ/ℤℓ(n)), where X̄ = X Xk k̄ for an algebraic closure k̄ of k. In this paper I study some properties of these Gk-modules; in particular, I am interested in the corank of the Galois cohomology groups

$${H^v}\,({G_k},{H^i}(\bar X,\,{Q_\ell }/{Z_\ell }(n))).$$

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Author information

Authors and Affiliations

  1. FB Mathematik, Universität Regensburg, Universitätstr. 31, 8400, Regensburg, Federal Republic of Germany

    Uwe Jannsen

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  1. Uwe Jannsen
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Editor information

Editors and Affiliations

  1. Faculty of Science, Department of Mathematics, Hongo, Tokyo 113, Japan

    Y. Ihara

  2. Department of Mathematics, University of California at Berkeley, 94720, Berkeley, CA, USA

    K. Ribet

  3. College de France, 3 Rue de Ulm, 75005, Paris, France

    J.-P. Serre

  4. Mathematical Sciences Research Institute, 1000 Centennial Drive, 94720, Berkeley, CA, USA

    K. Ribet

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Jannsen, U. (1989). On the ℓ-adic cohomology of varieties over number fields and its Galois cohomology. In: Ihara, Y., Ribet, K., Serre, JP. (eds) Galois Groups over ℚ. Mathematical Sciences Research Institute Publications, vol 16. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9649-9_5

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  • DOI: https://doi.org/10.1007/978-1-4613-9649-9_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9651-2

  • Online ISBN: 978-1-4613-9649-9

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