About the Bloch and Kato conjectures: Galois cohomology and values of \(L\)-functions. (Autour des conjectures de Bloch et Kato: Cohomologie galoisienne et valeurs de fonctions \(L\).) (French) Zbl 0821.14013

Fontaine, Jean-Marc; Perrin-Riou, Bernadette

Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 1, 599-706 (1994).

This article is of expository nature and explains the authors’ generalisation of the Bloch-Kato theory. The first section explains various local \(l\)-adic cohomologies, the second the globalisation, and the third the conjectures on \(L\)-functions and Tamagawa numbers.
For the entire collection see [Zbl 0788.00053].

Reviewer: G.Faltings (Bonn)

MSC:

14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11R34	Galois cohomology
14F30	\(p\)-adic cohomology, crystalline cohomology
19F27	Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11S25	Galois cohomology
14A20	Generalizations (algebraic spaces, stacks)

Keywords:

\(l\)-adic cohomologies; \(L\)-functions; Tamagawa numbers; Bloch-Kato conjecture