On Iwasawa theory of crystalline representations (Q1580257)

\textit{J.-M. Fontaine} [The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 249-309 (1990; Zbl 0743.11066)] obtained a general approach to the classification of \(p\)-adic representations of local fields. His method is based on the relation between local fields of characteristic \(0\) and functional local fields, given by the field of norms functor. As an application he proved that the reciprocity law of \textit{S. Bloch} and \textit{K. Kato} [The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 333-400 (1990; Zbl 0768.14001)] can be deduced from the explicit formula for Witt pairing in characteristic \(p\). In the paper under review the author applies the theory of Fontaine to Iwasawa theory of crystalline representations. In particular, for crystalline representations of finite height the explicit reciprocity law conjectured by \textit{B. Perrin-Riou} [Invent. Math. 115, 81-149 (1994; Zbl 0838.11071)] is proved. There were some conjectures in order to obtain a complete result. One of them is the explicit reciprocity law that makes it possible to study \(p\)-adic \(L\) functions at negative integers. The conjecture has already been proved by \textit{K. Kato, M. Kurihara} and \textit{T. Tsuji} [Local Iwasawa theory of Perrin-Riou and syntomic complexes, preprint 1996] and by \textit{P. Colmez} [Ann. Math. (2) 148, 485-571 (1998; Zbl 0928.11045)]. This third proof given by the author uses computation of Galois cohomology of \(p\)-adic representations using complexes of \(\Gamma\Phi\)-modules and classification of crystalline representations of finite height in terms of \(\Gamma\Phi\)-modules. By a result of \textit{P. Colmez} [J. Reine Angew. Math. 514, 119-143 (1999; Zbl 1191.11032)] we have that all crystalline representations of \(G_F\), the absolute Galois group of a finite unramified extension \(F\) of \(\mathbb{Q}_p\), are of finite height. The paper includes a description of crystalline representations in terms of \(\Gamma\Phi\)-modules, rings of \(p\)-adic representations, the explicit reciprocity law of Coleman, Kummer's homomorphism and the explicit reciprocity law of Perrin-Riou.