Explicit reciprocity laws for Hilbert symbols over higher-dimensional local fields at the prime 2

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 $\mathbb{Q}_p$; 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 $\mathbb{F}_q$. Typical mixed-characteristic examples are fraction fields of p-adic completions of power-series rings, denoted $\mathbb{Q}_p\{\{T_1\}\}\cdots\{\{T_{d-1}\}\}$. 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 $\theta_K\colon K_d^{\mathrm{top}}(K)\to\mathrm{Gal}(K^{\mathrm{ab}}/K)$, where $K_d^{\mathrm{top}}(K)$ 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 $\{a,1-a\}$ 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 $\mu_{p^n}$ of $p^n$-th roots of unity, the (d+1)-fold Hilbert symbol (norm residue symbol) $(\cdot,\ldots,\cdot)_{p^n}\colon(K^\ast)^{d+1}\to\mu_{p^n}$ is defined by applying $\theta_K$ to the first d arguments and evaluating the result on a chosen $p^n$-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 $K_d^{\mathrm{top}}(K)$ 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 $p>d+1$ (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 $p>2$; 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 $p\neq 2$. At $p=2$, explicit formulas are known only for one-dimensional local fields; nothing exists in dimension $\geq 2$.

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 $d\geq 2$ let $K$ be a d-dimensional mixed-characteristic local field of characteristic 0 whose first residue field has characteristic 2 (for example a finite extension of $\mathbb{Q}_2\{\{T_1\}\}\cdots\{\{T_{d-1}\}\}$), and assume $K$ contains the group $\mu_{2^n}$ of $2^n$-th roots of unity.

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

(2) For a Lubin–Tate formal group $F$ over a finite extension of $\mathbb{Q}_2$ contained in $K$ — 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 $K_d^{\mathrm{top}}(K)\times F(\mathfrak{m}_K)\to\kappa_{2^n}$, where $\mathfrak{m}_K$ is the maximal ideal of the ring of integers of $K$ and $\kappa_{2^n}$ is the group of $2^n$-torsion points of $F$ (assumed rational over $K$, as in the standard setups for odd $p$), at the same standard of explicitness achieved for odd $p$ 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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