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Explicit formulae for the Hilbert symbol of a formal group over the Witt vectors. (English. Russian original) Zbl 0889.11041
Izv. Math. 61, No. 3, 463-515 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 3, 3-56 (1997).
Let \(G\) be a formal group of finite dimension and finite height over an absolutely unramified complete discrete valuation ring \(o\) of characteristic 0 with perfect residue field \(k\) of characteristic \(p\). Let \(K\) be a finite extension of the field of fractions of \(o\) which contains all \(p^n\)-torsion points \(\kappa_n\) of \(G\). Denote by \(m\) the maximal ideal of the ring of integers \(O\) of \(K\). The standard pairing \(G_K\times G(m) \to \kappa_n\) is defined by the formula \((\sigma,f)=\sigma(h)-_G h\) where \([p^n]_G(h)=f\). If \(k\) is not separably \(p\)-closed (quasifinite), one can apply abelian \(p\)-class field theory of the reviewer [I. B. Fesenko, Russ. Acad. Sci., Izv. Math. 43, No. 1, 65–81 (1994; translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 4, 72–91 (1993; Zbl 0826.11056)] (classical class field theory resp.) to extend the pairing to the generalized Hilbert symbol between \(O^*\) (\(K^*\) resp.) and \(G(m)\).
Explicit formulae for the generalized Hilbert theorem have a very long history. In the context of the paper under review, D. G. Benois and S. V. Vostokov have constructed these in the case of \(n=1\) and formal groups in one parameter in [Algebra Anal. 2, 69-97 (1990; Zbl 0724.11059)].
In the paper under review the author generalizes his previous work on explicit formulae of Vostokov-Brückner’s type (multiplicative case) [V. A. Abrashkin, Math. Ann. 308, No. 1, 5–19 (1997; Zbl 0895.11049)] to construct explicit formulae for the generalized Hilbert symbol in the finite residue field case. The method originates from J.-M. Fontaine’s paper [Invent. Math. 115, No. 1, 151–161 (1994; Zbl 0802.14010)], in which Fontaine-Wintenberger’s theory of fields of norms was applied to cyclotomic extensions to relate Coates–Wiles’ type of explicit formulae to Witt’s pairing in characteristic \(p\). In the case of Vostokov-Brückner’s type, one uses instead the arithmetically profinite extension \(K'/K\) where \(K'=\cup K_i\), \(K_i=K_{i-1}(\pi_i)\), \(\pi_i^p=\pi_{i-1}\), \(\pi_0\) is a prime element of \(K_0=K\). The author applies Fontaine’s theory of \(p\)-adic periods of formal groups and uses an auxiliary construction of crystallographic symbols. As the result of virtuous technical calculations, two formulae for the generalized Hilbert pairing are deduced.

MSC:
11S31 Class field theory; \(p\)-adic formal groups
14L05 Formal groups, \(p\)-divisible groups
14F30 \(p\)-adic cohomology, crystalline cohomology
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