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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 continuoussection of the natural map from the arithmetic fundamental group of X to theabsolute Galois group of k. We discuss the algebraicity of this class in thecase of curves over p-adic fields, and deduce in particular a new proof ofStix's theorem according to which the index of a curve X over a p-adic field kmust be a power of p as soon as the natural map from the arithmetic fundamentalgroup of X to the absolute Galois group of k admits a section. Finally, anetale adaptation of Beilinson's geometrization of the pronilpotent completionof the topological fundamental group allows us to lift this cycle class insuitable cohomology groups.

Remarks on cycle classes of sections of the arithmetic fundamental group