Remarks on cycle classes of sections of the arithmetic fundamental group. (English) Zbl 1178.14019

Esnault, Hélène; Wittenberg, Olivier

Mosc. Math. J. 9, No. 3, 451-467 (2009).

Summary: Given a smooth and separated \(K(\pi ,1)\) variety \(X\) over a field \(k\), we associate a ”cycle class” in étale 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. Finally, an étale 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.

MSC:

14F35	Homotopy theory and fundamental groups in algebraic geometry
14C25	Algebraic cycles
14F20	Étale and other Grothendieck topologies and (co)homologies

Keywords:

étale fundamental group; cycle class map; pronilpotent completion