Weights in arithmetic geometry

Details Weights in arithmetic geometry Weights in arithmetic geometry

2010-03-03

arxiv.org/abs/1003.0927v1

Mathematics

Algebraic Geometry

27 pages

Scientific paper

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.

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