Explicit reciprocity laws for \(p\)-divisible groups over higher dimensional local fields (Q2703578)

The author extends the explicit reciprocity law given by \textit{D. Benois} for \(p\)-divisible groups over local fields with finite residue field [ J. Reine Angew. Math. 493, 115-151 (1997; Zbl 1011.11078)] and proves an explicit formula for the generalized Hilbert pairing associated to a \(p\)-divisible group over the ring of integers of an \((r+1)\)-dimensional field of mixed characteristic. NEWLINENEWLINENEWLINEMore precisely let \(L\) be an \((r+1)\)-dimensional local field of characteristic 0 whose first residue field \(k_r\) has characteristic \(p\) ( such that there exists a finite field \(k_0\) and a sequence of complete discrete valuation fields \(k_1,\dots, k_{r+1}=L\), with \(k_i\) the residue field of \(k_{i+1}\)) and let \(\overline L\) be the algebraic closure of \(L\). Let \(G\) be a \(p\)-divisible group over the ring of integers \(\mathcal O_L\) of dimension \(d\) and finite height \(h\) such that the \(p^n\)-torsion points \({}_{p^n}G( \mathcal O_{\overline L})\) over the integral closure \(\mathcal O_{\overline L}\) of \(\mathcal O_L\) are in \(L\). The generalized Hilbert pairing NEWLINE\[NEWLINEK_{r+1}^M(L) \times G(\mathfrak M_L) \rightarrow {}_{p^n}G (\mathcal O_L),NEWLINE\]NEWLINE where \(K_{r+1}^M(L)\) is the Milnor \(K\)-group, is defined in the standard way by the formula : \((\alpha,\beta)_{G,n}\) \(= \gamma ^{\rho_L( \alpha)} -_G \gamma\), where \(\gamma\) is a root of the equation \([p^n] \gamma = \beta\) and \(\rho_L \mid K_{r+1}^M(L) \rightarrow {\text{Gal}}(\overline L/ L)^{\text{ab}}\) is the reciprocity map in higher-dimensional local class field theory of \textit{K. Kato} [J. Fac. Sci. Univ. Tokyo, Sec. IA 26, 303-376 (1979; Zbl 0428.12013); ibid. 27, 603-683 (1980; Zbl 0463.12006); ibid. 29, 31-43 (1982; Zbl 0503.12004)]. NEWLINENEWLINENEWLINEIn the paper under review, the author uses Fontaine's interpretation of the Hodge-Tate theory and, in the spirit of the previous results of Artin-Hasse, Iwasawa, Wiles, Sen, Benois, etc., he proves for sufficient small values of \(\beta\) the following explicit formula NEWLINE\[NEWLINE(\alpha,\beta)_{G,n} = (-1)^r \operatorname {Res}_{L/ \mathbb Q_p} (\Psi_L (d\log(\alpha) \otimes \log_G(\beta)),NEWLINE\]NEWLINE where the crucial morphism \(\Psi_L\) is obtained via the theory of \textit{J.-M. Fontaine} on differential forms over local fields [Invent. Math. 65, 379-409 (1982; Zbl 0502.14015)].