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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.

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