, Volume 5, Issue 1, pp 73-102
Date: 02 Apr 2010

Weights in arithmetic geometry

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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.

This article is based on the 7th Takagi Lectures that the author delivered at the University of Tokyo on November 21–23, 2009.
Communicated by: Takeshi Saito