## 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 This construction, which is an unpublished result of Fontaine, can be considered as a vast generalization of Coleman’s isomorphism; we pause a little to explain this.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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### References:

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