For odd primes p, the higher-dimensional Hilbert symbol on mixed-characteristic d-dimensional local fields — and its analogs for Lubin–Tate, polynomial, and arbitrary one-dimensional formal modules — admits complete explicit formulas: the Vostokov school's power-series formulas on the higher Shafarevich basis, Kato's cohomological proof, Abrashkin–Jenni's field-of-norms reduction of the mixed-characteristic symbol to the characteristic-p Artin–Schreier–Witt formula, and Flórez's laws for arbitrary formal groups. All of these constructions assume p ≠ 2, and no explicit formula is known for the higher-dimensional symbol, or for any formal-module pairing over a higher local field, when p = 2 (the one-dimensional case at p = 2 is classical). The problem is to construct explicit reciprocity laws at p = 2 in dimension ≥ 2: closed-form formulas for the (d+1)-fold Hilbert symbol and for the Kummer pairings on formal modules over d-dimensional 2-adic local fields, matching the standard of explicitness and validity on the arguments achieved for odd primes.

Background

A one-dimensional local field is a complete discrete valuation field with finite residue field, for example a finite extension of the p-adic numbers Qp\mathbb{Q}_p; a d-dimensional local field is a complete discrete valuation field whose residue field is a (d−1)-dimensional local field, so that the successive residue fields end at a finite field Fq\mathbb{F}_q. Typical mixed-characteristic examples are fraction fields of p-adic completions of power-series rings, denoted Qp{{T1}}⋯{{Td−1}}\mathbb{Q}_p\{\{T_1\}\}\cdots\{\{T_{d-1}\}\}. Local class field theory describes the finite abelian extensions of such fields through a reciprocity map. Parshin (in characteristic p) and Kato (in general) constructed, for a d-dimensional local field K, a reciprocity homomorphism θK ⁣:Kdtop(K)→Gal(Kab/K)\theta_K\colon K_d^{\mathrm{top}}(K)\to\mathrm{Gal}(K^{\mathrm{ab}}/K), where Kdtop(K)K_d^{\mathrm{top}}(K) is the d-th topological Milnor K-group — the usual Milnor K-group of K modulo the closure of the subgroup generated by elements of the form {a,1−a}\{a,1-a\} with both a and 1−a in the ring of integers, a quotient adapted to the topology of K. When K has characteristic 0 and contains the group μpn\mu_{p^n} of pnp^n-th roots of unity, the (d+1)-fold Hilbert symbol (norm residue symbol) (⋅,…,⋅)pn ⁣:(K∗)d+1→μpn(\cdot,\ldots,\cdot)_{p^n}\colon(K^\ast)^{d+1}\to\mu_{p^n} is defined by applying θK\theta_K to the first d arguments and evaluating the result on a chosen pnp^n-th root of the last argument. An explicit reciprocity law is a formula computing this symbol, or the analogous Kummer pairing on a formal module, directly from the power-series expansions of the arguments in a fixed system of local parameters — without invoking class field theory. For formal modules one takes a Lubin–Tate formal group F over a one-dimensional local subfield k of K, i.e. a one-dimensional commutative formal group law whose endomorphism ring is the ring of integers of k and whose torsion points generate the totally ramified abelian extensions of k, and pairs Kdtop(K)K_d^{\mathrm{top}}(K) with the points of F on the maximal ideal of the ring of integers. The classical one-dimensional formulas (of Artin–Hasse, Iwasawa, and Vostokov type; see Vostokov's survey) express the pairing through residues, traces, and p-adic logarithms, and are proved on a Shafarevich basis — an explicit system of topological generators of the units built from primary elements, i.e. elements whose logarithms land in prescribed submodules — extending by bilinearity. In higher dimension the state of the art for odd p is as follows (surveys: Vostokov 2000; Benois 2025). Vostokov generalized his formula to mixed-characteristic higher local fields, the coincidence with the Galois-theoretic symbol being proved on the higher-dimensional analog of the Shafarevich basis; his 2013 paper rederives one-dimensional formulas via Witt's idea of primary elements (elements of the ring of p-typical Witt vectors — a characteristic-0 ring with functorial Frobenius lifting any perfect characteristic-p ring — detecting Brauer classes) and obtains formulas for a complete higher-dimensional field with perfect residue field. Kato gave a cohomological proof via syntomic complexes under the assumption p>d+1p>d+1 (as recounted in Benois's survey). Abrashkin and Jenni used the field-of-norms functor — Fontaine's construction attaching to a strictly deeply ramified tower of mixed-characteristic fields a characteristic-p local field whose Galois theory captures the tower — to deduce explicit formulas for the mixed-characteristic M-th Hilbert symbol on higher local fields from the Artin–Schreier–Witt (characteristic-p Witt symbol) formula, recovering Vostokov's formula as a special case. For formal modules over higher local fields, Afanas'eva–Bekker–Vostokov and Afanas'eva treated Lubin–Tate groups for both possibilities for the penultimate residue field (the last-but-one residue field in the chain) for p>2p>2; the Vostokov–Volkov series treated polynomial formal modules; Flórez constructed laws for the Kummer pairing of an arbitrary one-dimensional formal group with no restriction on the formal-module argument but conditions on the Milnor-K argument, while Fukaya's laws for p-divisible groups (connected p-power-torsion group schemes) are unrestricted in the K-theory argument but restrict the formal-module argument. Every one of these higher-dimensional constructions assumes p≠2p\neq 2. At p=2p=2, explicit formulas are known only for one-dimensional local fields; nothing exists in dimension ≥2\geq 2.

Problem Statement

Complete the explicit reciprocity program for higher-dimensional local fields at the prime 2, to the standard achieved for odd p. Concretely, for d≥2d\geq 2 let KK be a d-dimensional mixed-characteristic local field of characteristic 0 whose first residue field has characteristic 2 (for example a finite extension of Q2{{T1}}⋯{{Td−1}}\mathbb{Q}_2\{\{T_1\}\}\cdots\{\{T_{d-1}\}\}), and assume KK contains the group μ2n\mu_{2^n} of 2n2^n-th roots of unity.

(1) Construct an explicit formula for the (d+1)(d+1)-fold Hilbert symbol (⋅,…,⋅)2n ⁣:(K∗)d+1→μ2n(\cdot,\ldots,\cdot)_{2^n}\colon(K^\ast)^{d+1}\to\mu_{2^n}: a closed-form expression in the power-series expansions of the arguments in a fixed system of local parameters of KK, using multidimensional residues, traces, and 2-adic logarithms or Witt-vector data, which provably coincides with the Galois-theoretic symbol on all of (K∗)d+1(K^\ast)^{d+1}, for arbitrary nn.

(2) For a Lubin–Tate formal group FF over a finite extension of Q2\mathbb{Q}_2 contained in KK — and, as a generalization matching the odd-prime results of Flórez type, for arbitrary one-dimensional formal groups — construct an explicit formula for the Kummer pairing Kdtop(K)×F(mK)→κ2nK_d^{\mathrm{top}}(K)\times F(\mathfrak{m}_K)\to\kappa_{2^n}, where mK\mathfrak{m}_K is the maximal ideal of the ring of integers of KK and κ2n\kappa_{2^n} is the group of 2n2^n-torsion points of FF (assumed rational over KK, as in the standard setups for odd pp), at the same standard of explicitness achieved for odd pp and with validity restrictions no stronger than those of the best analogous odd-prime results.

These deliverables jointly serve one objective — explicit computational control of the reciprocity map of higher-dimensional local fields at the last untreated prime.

References

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