## Bloch and Kato’s exponential map: three explicit formulas.(English)Zbl 1064.11077

Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$, $$G_K$$ its absolute Galois group. In his article of 1993 on the approach to Iwasawa theory of $$L$$-functions via $${\mathbf B}_{dR}$$, K. Kato expressed his conviction that, for a de Rham representation $$V$$ of $$G_K$$, there should exist some “explicit reciprocity law”, namely “some explicit description of the relationship between $${\mathbf D}_{dR}(V)$$ and the Galois cohomology of $$V$$, or more precisely, some explicit descriptions of the maps $$\exp$$ and $$\exp^*$$ of $$V$$”. Recall that $$\exp_{K,V}:{\mathbf D}_{dR}(V)/\text{Fil}^\circ{\mathbf D}_{dR}(V)\to H^1(K, V)$$ is the Bloch-Kato exponential map and $$\exp^*_{K,V^*(1)}: H^1(K,V)\to\text{Fil}^\circ{\mathbf D}_{dR}(V)$$ the dual exponential map. For an unramified extension $$K/\mathbb{Q}_p$$ and a crystalline representation $$V$$, B. Perrin-Riou gave a precise conjectural reciprocity law using her “period map”, which is a vast generalization of Coleman’s map and an important ingredient in the construction of $$p$$-adic $$L$$-functions [Invent. Math. 115, 81–161 (1994; Zbl 0838.11071)]. Roughly speaking, Perrin-Riou constructed a family of maps $$\Omega_{V(j),h}$$, $$j\in\mathbb{Z}$$, $$h\in\mathbb{N}$$ which interpolates the exponential maps $$\exp_{K_n, V(j)}$$ at all the levels $$K_n$$ of the cyclotomic $$\mathbb{Z}_p$$-extension of $$K$$, and she conjectured an explicit formula expressing the cup product between $$\Omega_{V,h}$$ and $$\Omega_{V^*(1), 1-h}$$. This conjecture was proved independently (and by different methods) by D. Benois, P. Colmez and Kato-Kurihara-Tsuji.
The goal of the present article is to give formulas for $$\exp_{K,V}$$, $$\exp^*_{K,V^*(1)}$$ and $$\Omega_{V,h}$$ (the “three explicit formulas” of the title) in terms of the $$(\varphi,\Gamma)$$-module associated to $$V$$ by Fontaine’s theory. As a corollary, the author recovers a theorem of Colmez which states that Perrin-Riou’s map interpolates the $$\exp^*_{K,V^*(1-k)}$$ as $$k$$ runs over the negative integers (this is equivalent to Perrin-Riou’s conjectured reciprocity law). The author stresses that his results are not really new and are mostly a reinterpretation of formulas of Benois, Colmez and Kato-Kurihara-Tsuji (for Perrin-Riou’s map) and of Cherbonnier-Colmez (for the dual exponential map) in the language of his article [L. Berger, Invent. Math. 14, No. 8, 219–284 (2002; Zbl 1113.14016)] on $$p$$-adic representations and differential equations. Nevertheless, the interpretation in terms of $$(\varphi,\Gamma)$$-modules gives a simplification and an improvement of the construction of the period maps $$\Omega_{V,h}$$ and, therefore, perhaps the most “natural” proof of Perrin-Riou’s reciprocity law. In addition, it should generalize to de Rham representations and to other settings than the cyclotomic one.

### MSC:

 11S25 Galois cohomology 11F80 Galois representations 11R23 Iwasawa theory 14F30 $$p$$-adic cohomology, crystalline cohomology

### Citations:

Zbl 1113.14016; Zbl 0838.11071
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