Extending K-theoretic flag-symbol reciprocity to arithmetic bases: realizing Hilbert reciprocity over $\operatorname{Spec}\mathbb{Z}$

Musicantov and Yom Din associated to each full flag $\mathcal{F}$ in an $n$ -dimensional variety $X$ over a field a spectrum-level symbol map $\mu_{\mathcal{F}}:K(F_X)\to\Sigma^n K(k)$ and proved that the sum of the symbols over all full flags refining a partial flag is zero, recovering the degree, Weil, residue, Contou-Carrère, and Parshin reciprocity laws on K-groups. They asked whether this setup can be altered so that the Hilbert reciprocity law for number fields fits in, which requires addressing the archimedean prime, wild ramification (notably at the prime $2$ ), and the role of the sphere spectrum underlying all primes. Later work realized the full Hilbert symbol, including its wild part, as a K-theory localization boundary map away from $2$ (Braunling's optimal orders), and produced exact Hilbert-reciprocity-type sequences from the K-theory of locally compact vector spaces, but no extension of the flag-symbol framework to arithmetic bases exists, the prime $2$ and the real places remain unmanaged in any boundary-map realization, and a non-circular K-theoretic derivation of Hilbert reciprocity is still missing.

Background

Algebraic K-theory associates to a scheme or ring $R$ a spectrum $K(R)$ (a sequence of pointed spaces bonding maps between them) whose homotopy groups $K_i(R)$ are the classical K-groups: roughly $K_0$ classifies vector bundles, $K_1$ captures units and elementary matrices, and $K_2$ records Steinberg-type relations. Working with the spectrum rather than only its homotopy groups keeps track of all $K_i$ and of the maps between them simultaneously; $\Sigma^n$ denotes $n$ -fold suspension of a spectrum, shifting homotopy groups. Let $X$ be an $n$ -dimensional irreducible variety over a field $k$ , with field of rational functions $F_X$ . A full flag $\mathcal{F}$ is a chain of closed irreducible subvarieties $X=X_0\supset X_1\supset\cdots\supset X_n$ with $\operatorname{codim}_X(X_i)=i$ . Musicantov and Yom Din constructed a symbol map of spectra $\mu_{\mathcal{F}}:K(F_X)\to\Sigma^n K(k)$ and proved an abstract reciprocity law: for a partial flag $G$ (a full flag with one term of codimension $0<d\le n$ omitted), the sum of $\mu_{\mathcal{F}}$ over all full flags $\mathcal{F}$ refining $G$ vanishes. Reading this on K-groups recovers, for curves, the vanishing of the degree of a principal divisor, Weil reciprocity, the residue theorem, and Contou-Carrère reciprocity (the Contou-Carrère symbol is the universal generalization of the tame symbol for the Laurent series field $k((t))$ ), and in higher dimension Parshin reciprocity. The tame symbol itself is the boundary map $\partial_v:K_2(F)\to\kappa(v)^\times$ of the K-theory localization sequence at a discrete valuation $v$ with residue field $\kappa(v)$ ; it sees only tamely ramified behavior. In number theory, every completion $F_v$ of a number field $F$ at a non-complex place carries the Hilbert symbol $h_v:K_2(F_v)\to\mu(F_v)$ (into the roots of unity), constructed via local class field theory, and the tame symbol captures exactly its prime-to- $p$ part at a $p$ -adic place; the wild part is invisible to the plain localization boundary. Hilbert reciprocity is the product formula $\prod_v h_v(\alpha,\beta)=1$ over all places $v$ of $F$ , the complex places contributing trivially; it is the number-field analogue of Weil reciprocity for curves over finite fields, which Gillet derived from K-theory localization. Under the function-field/number-field dictionary one would want $X=\operatorname{Spec}\mathcal{O}_F$ or $\operatorname{Spec}\mathbb{Z}$ to play the role of the curve, and Musicantov and Yom Din explicitly asked whether their setup can be altered accordingly. They point out three obstructions: the archimedean prime (a real place is not a discrete valuation ring), wild ramification, especially the behavior at the prime $2$ , and the role of the sphere spectrum (the initial object among spectra, which in homotopy theory underlies all primes at once, as in Clausen's p-adic J-homomorphisms — analogues for $p$ -adic vector spaces of the classical real J-homomorphism arising from one-point compactification of real vector spaces — whose product formula specializes on $\pi_2$ to Hilbert's quadratic reciprocity law).

Problem Statement

Alter the Musicantov–Yom Din reciprocity framework — symbol maps $\mu_{\mathcal{F}}:K(F_X)\to\Sigma^n K(k)$ attached to full flags, with the reciprocity law that the sum of symbols over full flags refining a partial flag vanishes — so that the Hilbert reciprocity law for number fields fits in as an instance. Concretely: construct a version of the flag-symbol machinery in which arithmetic bases such as $\operatorname{Spec}(\mathbb{Z})$ or $\operatorname{Spec}(\mathcal{O}_F)$ play the role of the variety over a field, define the local symbol maps, and prove a reciprocity statement for them that is Hilbert reciprocity for $F$ — $F$ an arbitrary number field — i.e. the product formula $\prod_v h_v(\alpha,\beta)=1$ over all places $v$ of $F$ (complex places contributing trivially), with the full Hilbert symbol $h_v$ including its wild part. The construction must address the three phenomena the source names: the archimedean prime, wild ramification including the behavior at the prime $2$ , and the role of the sphere spectrum underlying all primes. Two components of the problem are of independent interest: realizing the full Hilbert symbol (all $p$ , including $2$ , and the real places) in a K-theoretic setting, and identifying the reciprocity target group with the roots of unity non-circularly. A solution phrased only up to $2$ -primary torsion, or only away from the prime $2$ and the real places, does not settle the question.

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