Japanese Journal of Mathematics

, Volume 5, Issue 1, pp 73–102

Weights in arithmetic geometry

Article

Abstract.

The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more recent applications of weights, e.g. to Hasse principles and the computation of motivic cohomology, and discuss some open questions.

Keywords and phrases:

weights étale cohomology Hasse principles 

Mathematics Subject Classification (2010):

11G25 11G35 14F20 14F30 

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Copyright information

© The Mathematical Society of Japan and Springer Japan 2010

Authors and Affiliations

  1. 1.NWF I-MathematikUniversität RegensburgRegensburgGermany

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