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Théorie d’Iwasawa des représentations \(p\)-adiques d’un corps local. (Iwasawa theory of \(p\)-adic representations of a local field). (French) Zbl 0933.11056
This paper contains, besides a lot of interesting technical material, two principal results. Firstly it exhibits, for a general \(\mathbb{Z}_p\)-representation or \(p\)-adic representation \(V\) of the absolute Galois group of a \(p\)-adic field \(K\), an isomorphism \(\text{Exp}^*_{V^*(1)}\) between the Iwasawa cohomology group \(H^1_{\text{Iw}}(K,V)\) and a certain “linear object” \(D(V)^{\psi=1}\) associated to \(V\) for the purpose. (Remark: As one will see, this map is much rather a logarithm than an exponential; the notation suggests that it is dual to an exponential.) This construction, which is an unpublished result of Fontaine, can be considered as a vast generalization of Coleman’s isomorphism; we pause a little to explain this.
One takes \(K=\mathbb{Q}_p\) and \(V=\mathbb{Z}_p(1)\). Then Coleman theory essentially calculates the projective limit lim\(_n K_n^*\), with \(K_n=\mathbb{Q}_p(\zeta_{p^n}\)) and the transition maps being the norm maps, in terms of a certain additive module of power series. The isomorphism connecting the two objects is a kind of logarithmic derivative; the projective limit in question is canonically isomorphic to the Iwasawa cohomology of \(V\); and the target of the isomorphism is, roughly, \(D(V)^{\psi=1}\). At any rate, \(\psi\) is, up to a scalar factor \(p^{-1}\), the same as Coleman’s trace operator \(\mathcal S\). The paper under review extends all this to arbitrary \(\mathbb{Z}_p\)-representation or \(p\)-adic representations; in particular it provides a generalization of Coleman theory to any \(p\)-adic base field \(K\), even ramified over \(\mathbb{Q}_p\).
As a second major result, the authors connect their map \(\text{Exp}^*_{V^*(1)}\) with the “exponentials” of Bloch and Kato, in case the representation \(V\) is de Rham. Their Theorem IV 2.1. constitutes an equality, where on the left hand we have an expression \(\text{Exp}^*_{V^*(1)}(\mu)\), and on the right hand a sum involving Bloch-Kato exponentials for the twists \(V(k)\), \(k\) running over all integers. This is in a way logical since Iwasawa cohomology does not distinguish \(V\) from its twists \(V(k)\).
Lastly, as a consequence, the relation between \(\text{Exp}^*_{V^*(1)}\) and Perrin-Riou’s logarithm is elucidated, in case \(K\) is unramified and the representation \(V\) is crystalline. Very roughly, getting from the latter to the former amounts to following a logarithm with a derivative, in other words: passing from a logarithm map to a logarithmic derivative.
The paper also contains a discussion of “overconvergent” \(p\)-adic representations; the authors have shown in another paper (to appear) that every representation of the absolute Galois group of \(K\) is overconvergent. The subject matter of the paper under review is complicated and partly very technical, but the writing style is clear, elegant, and witty.
(Just one rather minor remark: it seems that for \(p=2\) the condition on page 245 ensuring that the cyclotomic extension \(K_\infty/K\) has procyclic Galois group \(\Gamma_K\) is not totally accurate; for \(K= \mathbb{Q}_2(\sqrt 2)\), \(\Gamma_K\) is not procyclic.).

MSC:
11S23 Integral representations
11R23 Iwasawa theory
11S25 Galois cohomology
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