zbMATH — the first resource for mathematics

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 This construction, which is an unpublished result of Fontaine, can be considered as a vast generalization of Coleman’s isomorphism; we pause a little to explain this.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

11S23 Integral representations
11R23 Iwasawa theory
11S25 Galois cohomology
Full Text: DOI
[1] Spencer Bloch and Kazuya Kato, \?-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333 – 400. · Zbl 0768.14001
[2] Cherbonnier F.: Représentations \(p\)-adiques surconvergentes, thèse de l’université d’Orsay, 1996.
[3] Cherbonnier F., Colmez P.: Représentations \(p\)-adiques surconvergentes, Inv. Math. (à paraître). · Zbl 0928.11051
[4] Robert F. Coleman, Division values in local fields, Invent. Math. 53 (1979), no. 2, 91 – 116. · Zbl 0429.12010 · doi:10.1007/BF01390028 · doi.org
[5] Colmez P.: Théorie d’Iwasawa des Représentations de de Rham d’un Corps Local, Ann. of Math. (à paraître). · Zbl 0928.11045
[6] Colmez P.: Représentations cristallines et représentations de hauteur finie, Prépublication du LMENS-97-28, 1997.
[7] Jean-Marc Fontaine and Jean-Pierre Wintenberger, Le ”corps des normes” de certaines extensions algébriques de corps locaux, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 6, A367 – A370 (French, with English summary). · Zbl 0475.12020
[8] Jean-Marc Fontaine, Représentations \?-adiques des corps locaux. I, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 249 – 309 (French). · Zbl 0743.11066
[9] Jean-Marc Fontaine, Le corps des périodes \?-adiques, Astérisque 223 (1994), 59 – 111 (French). With an appendix by Pierre Colmez; Périodes \?-adiques (Bures-sur-Yvette, 1988). · Zbl 0940.14012
[10] Jean-Marc Fontaine, Représentations \?-adiques semi-stables, Astérisque 223 (1994), 113 – 184 (French). With an appendix by Pierre Colmez; Périodes \?-adiques (Bures-sur-Yvette, 1988). · Zbl 0865.14009
[11] Herr L.: Cohomologie Galoisienne des corps \(p\)-adiques, thèse de l’université d’Orsay, 1995.
[12] Kazuya Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil \?-functions via \?_\?\?. I, Arithmetic algebraic geometry (Trento, 1991) Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, pp. 50 – 163. · Zbl 0815.11051 · doi:10.1007/BFb0084729 · doi.org
[13] Bernadette Perrin-Riou, Théorie d’Iwasawa des représentations \?-adiques sur un corps local, Invent. Math. 115 (1994), no. 1, 81 – 161 (French). With an appendix by Jean-Marc Fontaine. · Zbl 0838.11071 · doi:10.1007/BF01231755 · doi.org
[14] Jean-Pierre Wintenberger, Le corps des normes de certaines extensions infinies de corps locaux; applications, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 1, 59 – 89 (French). · Zbl 0516.12015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.