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Iwasawa theory of de Rham representations of a local field. (Théorie d’Iwasawa des représentations de de Rham d’un corps local.) (French) Zbl 0928.11045
In her seminal paper on Iwasawa theory of local $$p$$-adic representations [Invent. Math., 115, No. 1, 81-149 (1994; Zbl 0838.11071)], B. Perrin-Riou considers generalized Coleman maps as machineries to produce generalized $$p$$-adic $$L$$-functions.
More specifically, the usual Coleman map associates to any norm-coherent system $$u$$ of local units in $$\bigcup_n \mathbb{Q}_n(\mu_{p^n})$$, a measure $$\lambda_u$$ on $$\Gamma= \text{Gal} (\mathbb{Q}_p (\mu_{p^\infty})/ \mathbb{Q}_p)$$ which is uniquely determined by a certain integration formula involving the logarithm of the well-known Coleman series and choosing $$u$$ to be a suitable system of cyclotomic units, the pseudo-measure $$(1-\gamma)^{-1}\lambda_u$$ (where $$\gamma$$ is a topological generator of $$\Gamma)$$ gives the Kubota-Leopoldt $$p$$-adic zeta function.
More generally, if $$V$$ is a $$p$$-adic crystalline representation of the absolute Galois group of a finite unramified extension $$K$$ of $$\mathbb{Q}_p$$, B. Perrin-Riou interpolates $$p$$-adically the Bloch-Kato exponential maps, in the sense that she constructs an exponential map $$\text{Exp}_{V,h}$$ $$(h\gg 0)$$ which is uniquely determined by a certain integration formula involving the Bloch-Kato maps $\exp_{V(k)}:D_{dR} \bigl(V(k) \bigr)\to H^1\bigl( K,V(k)\bigr),\qquad k\geq 1-h.$ For $$V=\mathbb{Q}_p(1)$$, one recovers the inverse of the Coleman map, and it is hoped that $$p$$-adic $$L$$-functions could be attached to crystalline representations $$V$$ (i.e. representations with “good reduction”) by using $$\text{Exp}_{V,h}$$ and compatible systems of elements in motivic cohomology.
One important conjecture proposed by B. Perrin-Riou in this process was a reciprocity law, $$\text{Rec}(V)$$, a far reaching generalization of the usual reciprocity law of local class-field theory (corresponding to $$V=\mathbb{Q}_p(1))$$, related to an eventual functional equation for $$p$$-adic $$L$$-functions. In the present paper, P. Colmez extends the construction of the exponential map to the base of “bad reduction”, i.e. to de Rham representations $$V$$ over a finite extension $$K$$ of $$\mathbb{Q}_p$$, using distributions of finite order in $$p$$-adic spaces.
For $$h\geq 1$$, he constructs a map $$\text{Exp}_V^{(p)}$$ which is uniquely determined by a certain integration formula involving the Bloch-Kato $$\exp_{V(-k)}$$, $$0\leq k\leq h-1$$. By Fourier transforms, he shows that, in the setting of Perrin-Riou, the two maps $$\text{Exp}^{(h)}_V$$ and $$\text{Exp}_{V,h}$$ coincide up to a normalization. His construction is explicit enough to give also a logarithm map $$\text{Log}_H^{(h)}$$ such that $$\text{Log}_V^{(h)} \circ\text{Exp}_V^{(h)}$$ is the identity modulo the kernel of $$\text{Exp}^{(h)}_V$$ (which is finite). He then shows a reciprocity law for the logarithm, i.e. a description of $$\text{Log}_V^{(h)}$$ in terms of the dual Bloch-Kato exponential $$\exp^*_{V^*(1+k)}$$, $$k\notin [0,h-1]$$. This yields Perrin-Riou’s conjecture $$\text{Rec}(V)$$ if $$V$$ is crystalline and $$K$$ unramified over $$\mathbb{Q}_p$$.
NB: The conjecture $$\text{Rec}(V)$$ has also been proved independently by Kato-Kurihara-Tsuj (resp. D. Benois) using syntomic cohomology (resp. Galois cohomology via $$(\varphi,\Gamma)$$-modules).

##### MSC:
 11R23 Iwasawa theory 14F30 $$p$$-adic cohomology, crystalline cohomology 11S20 Galois theory 11S25 Galois cohomology
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