Remarks on cycle classes of sections of the arithmetic fundamental group

Details Remarks on cycle classes of sections of the arithmetic fundamental group Remarks on cycle classes of sections of the arithmetic fundamental group

2008-07-18

arxiv.org/abs/0807.2963v4

Mathematics

Algebraic Geometry

19 pages; final version (added MSC codes and keywords)

Scientific paper

Given a smooth and separated K(pi,1) variety X over a field k, we associate a "cycle class" in etale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute Galois group of k. We discuss the algebraicity of this class in the case of curves over p-adic fields, and deduce in particular a new proof of Stix's theorem according to which the index of a curve X over a p-adic field k must be a power of p as soon as the natural map from the arithmetic fundamental group of X to the absolute Galois group of k admits a section. Finally, an etale adaptation of Beilinson's geometrization of the pronilpotent completion of the topological fundamental group allows us to lift this cycle class in suitable cohomology groups.

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