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.


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