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Generalized explicit reciprocity laws. (English) Zbl 1024.11029
The classical explicit reciprocity law (established by Artin, Hasse, Iwasawa et al.) is the theory to give explicit descriptions of Hilbert symbols. Let \(K_n=K(\zeta_{p^n})\), where \(K\) is a finite unramified extension of \(\mathbb{Q}_p\). Consider, for \(m\geq 0\), a homomorphism \[ \lambda_m: \varprojlim_n (K_n)^\times\to K_m \] defined by using the theory of Hilbert symbols. The classical explicit reciprocity law is concerned with the explicit description of this homomorphism in terms of differential forms.
The author generalizes the explicit reciprocity law to \(p\)-adically complete discrete valuation fields whose residue fields are not necessarily perfect. The ring \(B_{dR}\) defined by J. M. Fontaine and G. Laffaille [Ann. Sci. Eć. Norm. Supér. (4) 115, 547-608 (1982; Zbl 0579.14037)] in the perfect residue field case, and generalized by G. Faltings [J. Am. Math. Soc. 1, 255-299 (1988; Zbl 0764.14012)] to the general case, plays a central role in this theory. To be more precise, let \((K, \Lambda,G)\) be a triple consisting of a complete discrete valuation field \(K\) of characteristic zero with residue field \(k\) of characteristic \(p>0\), \(G\) a one-dimensional \(p\)-divisible group over the ring of integers \({\mathcal O}_K\), and \(\Lambda\) an integral domain over \(\mathbb{Z}_p\) which is free of finite rank as a \(\mathbb{Z}_p\)-module, endowed with an injective homomorphism \(\Lambda\to\text{End}(G)\). One assumes further that the Tate module \(T_pG\) is a free \(\Lambda\)-module and a certain \(K\)-map induced by a connection on \(D_{dR}(V_pG)\) is bijective (condition (iii)). Let \(h=\text{rank}_\Lambda T_pG\). In the generalized explicit reciprocity law one replaces \(\varprojlim_nK_n^\times\) by the inverse limit \(\varprojlim_nK_h^M(Y_n)\), (with respect to norm maps, where \(Y_n\) is a certain finite etale scheme over \(\text{Spec}(K)\) and \(K_h^M\) denotes Milnor’s \(K\)-theory) and one replaces \(\lambda_m\) by \[ \lambda^s_m: \varprojlim_n K_h^M(Y_n)\to{\mathcal O}(Y_m) \otimes_{{\mathcal O}_K}\text{coLie}(G)^{\otimes (h+r)}, \] \((s=(s(i))_{1\leq i\leq h}\in\mathbb{N}^h\), \(r=\sum^h_{i=1}s(i))\) which are defined by using \(B_{dR}\) and Galois cohomology. The generalized explicit reciprocity law is a statement concerning the description of \(\lambda^s_m\) in terms of differential forms.
The results of this article will apply to \(p\)-adic completions of the function fields of modular curves, whose residue fields are function fields in one variable over finite fields [K. Kato, \(p\)-adic Hodge theory and special values of zeta functions of elliptic cusp forms, preprint (2000)].

11F33 Congruences for modular and \(p\)-adic modular forms
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
11R23 Iwasawa theory
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture