The residue-characteristic-2 case of Shafarevich-style explicit formulas for the Hilbert pairing on complete higher-dimensional fields

S. V. Vostokov's 2013 Mat. Sb. paper on Shafarevich's 'A general reciprocity law' develops a method based on Witt's identification of the Brauer group of a field with its group of primary elements and proves explicit formulas for the Hilbert symbol in a classical local field (Theorem 1) and for the Hilbert pairing on a complete $n$-dimensional field with perfect residue field (Theorems 2-4) — in every case only for odd $p$; the value is expressed as a power of a root of unity with exponent $a_\varphi=A^\varphi-A$, where $A^\Delta-A=a$, but $a_\varphi$ is determined only modulo $(p^m,\wp)$. The open problem addressed here is the residue-characteristic-2 case in the higher-dimensional setting: construct, with complete proof, an explicit and effectively computable formula for the Hilbert pairing on complete higher-dimensional fields containing the $2^m$-th roots of unity. The one-dimensional case is known by other methods (Henniart; Vostokov-Fesenko; Kölcze; Belyaeva-Vostokov); for higher fields only an announced, incompletely published $p=2$ construction exists (Beliaeva-Vostokov 1998/2001), and all subsequent approaches work with $p$ odd.

Background

Explicit reciprocity laws are closed-form expressions for the norm-residue (Hilbert) symbol, the pairing that encodes local class field theory. For a field $K$ that is a finite extension of $\mathbb{Q}_p$ and contains the group $\mu_{p^n}$ of $p^n$-th roots of unity, the Hilbert symbol is defined by $(\alpha,\beta)=\Psi_K(\alpha)(\sqrt[p^n]{\beta})/\sqrt[p^n]{\beta}$, where $\Psi_K:K^*\to\operatorname{Gal}(K^{ab}/K)$ is the local reciprocity map of class field theory; computing it explicitly is regarded as the final step of Hilbert's ninth problem on the most general reciprocity law. I. R. Shafarevich's 1950 paper 'A general reciprocity law' introduced the technique of primary elements — an element $\omega\in K^*$ is $p^n$-primary if the extension $K(\sqrt[p^n]{\omega})/K$ is unramified — together with explicit generating systems of $K^*$ now called Shafarevich bases, and these ideas underlie most explicit formulas, notably S. V. Vostokov's 1978 formula expressing $(\alpha,\beta)$ through the residue of a power series built from the Artin-Hasse function $E(X)=\exp(X+X^p/p+X^{p^2}/p^2+\cdots)$ and the logarithms of the arguments.

The higher-dimensional setting is as follows. A complete $n$-dimensional field is a field complete with respect to a discrete valuation whose residue field is a complete $(n-1)$-dimensional field, the chain ending in a perfect field $k$ of characteristic $p>0$; when $k$ is finite this is a higher local field in the usual sense. For such fields A. N. Parshin and K. Kato constructed local class field theory on the topological Milnor $K$-groups $K_n^{top}(K)$ (Milnor $K$-theory endowed with the topology in which valuation-open unit subgroups generate the topological group), and I. B. Fesenko extended the theory to complete fields with an arbitrary perfect residue field. Writing $W(k)$ for the ring of Witt vectors (infinite sequences of residue-field elements with carry-wise addition, on which the Frobenius $\Delta$ acts coordinate-wise) and $\wp(x)=x^\Delta-x$, the theory requires $\kappa=\dim_{\mathbb{F}_p}k/\wp(k)>0$, i.e. the residue field admits a nontrivial Artin-Schreier extension. For $K$ containing $\mu_{p^m}$ one then has the Hilbert pairing

$$(\cdot,\cdot):\ K_n^{top}(K)\times K^*/K^{*p^m}\longrightarrow\mu_{p^m},$$

defined through Fesenko's reciprocity map $\Psi_K:VK_n^{top}(K)\to\operatorname{Hom}_{\mathbb{Z}_p}(\operatorname{Gal}(K^{ab,p}/K^{nr,p}),\mu_{p^m})$, where $K^{nr,p}=K\cdot W(k^{ab,p})$ is the maximal abelian unramified $p$-extension of $K$ and $VK_n^{top}(K)$ is the subgroup generated by principal units (units congruent to 1 modulo the maximal ideal); evaluating at each $\varphi\in\operatorname{Gal}(K^{nr,p}/K)$ yields the $\mu_{p^m}$-valued pairing written informally in the source and used in the Problem Statement below.

Vostokov's 2013 paper, dedicated to Shafarevich's memory, returns to this circle of ideas with a new method based on Witt's observation that the Brauer group of a field containing $\mu_{p^m}$ identifies with the group of $p^m$-primary elements. It proves the classical local-field formula (Theorem 1) and the higher-dimensional formula (Theorems 2-4): for local parameters $t_1,\dots,t_n$ and arguments $\alpha_1,\dots,\alpha_{n+1}\in K^*$,

$$(\{\alpha_1,\dots,\alpha_n\},\alpha_{n+1})(\varphi)=\zeta^{a_\varphi},\qquad\varphi\in\operatorname{Gal}(K^{nr,p}/K),$$

where $\zeta$ is a primitive $p^m$-th root of unity, $a$ is a multidimensional residue of an explicitly written differential expression $\Phi(\alpha_1,\dots,\alpha_{n+1})/(\zeta^{p^m}-1)$ built from logarithms and logarithmic derivatives of the power-series expansions of the $\alpha_i$ in $t_1,\dots,t_n$, and $a_\varphi=A^\varphi-A$ for a solution $A$ of the Artin-Schreier-type equation $A^\Delta-A=a$. Every theorem is proved under the hypothesis $p\neq 2$: the two congruence lemmas carrying the computation (Lemma 3 and the main Lemma 9, comparing the truncated logarithm $l(\varepsilon)$ with $(1-\Delta)\log\varepsilon$ modulo $p^n$) are stated and proved only for $p>3$, and Remark 5 notes that $a_\varphi$ is determined only modulo $(p^m,\wp)$ — the element $A$ exists and $a_\varphi$ is independent of its choice, but $A$ is not described explicitly.

For one-dimensional local fields the $p=2$ explicit formulas are classical: G. Henniart (1981) and Vostokov-Fesenko (1984-85, Lubin-Tate setting) obtained them, P. Kölcze (1995) proved the Brückner-Vostokov series formula is valid for $p=2$, and Belyaeva-Vostokov worked out the detailed one-dimensional theory for arbitrary $p$, odd and even, including the correction terms (adding $1/2$ to the term $1/(\zeta^{p^n}-1)$) that the 2013 paper itself cites. For higher fields, however, every complete published formula assumes $p$ odd: Vostokov's 1985 multidimensional construction, the Lubin-Tate generalizations of Afanas'eva-Bekker-Vostokov and Afanas'eva (2013-14, residue characteristic $p>2$), Zinov'ev's Artin-Hasse and Iwasawa type formulas (2002-06), Abrashkin-Jenni's field-of-norms derivation (2012, from the characteristic-$p$ Witt-symbol formula for $p>2$), and Flórez's norm-residue symbol (2017, $p>2$). The only $p=2$ higher-dimensional line is an announced construction of Beliaeva-Vostokov (1998; 2001 Part I) whose published proof of the crucial symbol property is incomplete, the complete proof being promised in a continuation that never appeared. This is the open problem addressed here: the effectively computable, completely proved residue-characteristic-2 formula in the higher-dimensional setting.

Problem Statement

Let $p=2$. Let $K$ be a complete $n$-dimensional field with $n\ge 2$, of characteristic 0, whose last residue field $k$ is perfect of characteristic 2 with $\kappa=\dim_{\mathbb{F}_2}k/\wp(k)>0$, and assume $\mu_{2^m}\subset K$, so that the Hilbert pairing

$$(\cdot,\cdot):\ K_n^{top}(K)\times K^*/K^{*2^m}\longrightarrow\mu_{2^m}$$

of Fesenko's higher local class field theory is defined, evaluated at each $\varphi\in\operatorname{Gal}(K^{nr,2}/K)$. Construct, with complete proof, an explicit formula for this pairing of the Shafarevich-Vostokov type — an effectively computable value $\zeta^{a_\varphi}$, where $a_\varphi$ is obtained from the arguments $\alpha_1,\dots,\alpha_{n+1}\in K^*$ through residues of explicitly constructed power series in their expansions in local parameters $t_1,\dots,t_n$ — in the spirit of Theorems 2-4 of Vostokov (2013), which cover only $p\neq 2$. The formula must be well defined — independent of the choices of series expansions, local parameters, prime element, and of the solution $A$ of $A^\Delta-A=a$ entering $a_\varphi=A^\varphi-A$ — and must coincide with the Hilbert pairing via the 2-primary reciprocity map. The formula is sought in the generality of the source's Theorems 3-4: complete higher-dimensional fields with an arbitrary perfect residue field of characteristic 2 satisfying $\kappa>0$, in the mixed-characteristic case; standard higher local fields (finite last residue field) form the particular case corresponding to the source's Corollary 4. The one-dimensional case is excluded because it is already known by other methods (Henniart; Vostokov-Fesenko; Kölcze).

References

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