## 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

### Citations:

Zbl 0724.11059; Zbl 0802.14010; Zbl 0895.11049; Zbl 0826.11056
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