Bloch and Kato’s exponential map: three explicit formulas.

*(English)*Zbl 1064.11077Let \(K\) 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.

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.

Reviewer: Thong Nguyen Quang Do (BesanĂ§on)

##### MSC:

11S25 | Galois cohomology |

11F80 | Galois representations |

11R23 | Iwasawa theory |

14F30 | \(p\)-adic cohomology, crystalline cohomology |