Realizing the full Hilbert symbol, including its wild part, as a boundary map of a K-theory localization sequence, and a localization-theoretic proof of Hilbert reciprocity for number fields

For a local field, the Hilbert symbol is the pairing linking local class field theory with Kummer theory; the boundary map of the K-theory localization sequence at a discrete valuation, the tame symbol, sees only its prime-to-p part. Gillet's proof of Weil reciprocity from the localization sequence yields Hilbert reciprocity for function fields, and it has been a long-standing folklore question whether the full Hilbert symbol, including its p-primary wild part, can itself be realized as a localization boundary, so that global Hilbert reciprocity for number fields would become the exactness of a K-theory localization sequence. Braunling (2021, published 2023) resolved this affirmatively at all finite places above odd primes, using singular 'optimal orders' obtained by pinching the maximal order near the ramification locus. The surviving open core is the 2-primary wild part at places above the prime 2, the contribution of real (archimedean) places, and a non-circular computation of $SK_1$ of the global order.

Background

A number field $F$ has completions $F_v$ at its places $v$: finite extensions of $\mathbb{Q}_p$ at finite primes $p$ (p-adic local fields), and $\mathbb{R}$ or $\mathbb{C}$ at the archimedean places. Write $\mu(F_v)$ for the finite group of roots of unity in $F_v$ and $m_v=\#\mu(F_v)$. The Hilbert symbol of a noncomplex local field is the pairing $h_v\colon F_v^{\times}\times F_v^{\times}\to\mu(F_v)$ defined through local class field theory - the classification of the abelian extensions of $F_v$, whose Artin reciprocity map $\mathrm{Art}_{F_v}\colon F_v^{\times}\to\mathrm{Gal}(F_v^{\mathrm{ab}}/F_v)$ identifies the multiplicative group with the Galois group of the maximal abelian extension - and Kummer theory (the study of field extensions obtained by adjoining roots of elements): for $y\in F_v^{\times}$ with a chosen $m_v$-th root $\sqrt[m_v]{y}$, one sets $h_v(x,y)=\mathrm{Art}_{F_v}(x)(\sqrt[m_v]{y})/\sqrt[m_v]{y}$. It is bilinear and satisfies the Steinberg relation $h_v(x,1-x)=1$, so it factors through $K_2(F_v)$, the second Milnor K-group ($K_2$ of a field is $F^{\times}\otimes F^{\times}$ modulo the subgroup generated by symbols $\{x,1-x\}$). Over a p-adic field, $\mu(F_v)$ splits into its prime-to-p part, which lifts from the residue field, and its p-primary part; the latter is governed by wild ramification and is called the wild part of the Hilbert symbol. Formulas for it (Artin–Hasse, Shafarevich, Wile, Vostokov, and others) constitute the theory of explicit reciprocity laws and are notoriously involved.

Algebraic K-theory supplies a different symbol. For a field $F$ with a discrete valuation $v$, valuation ring $\mathcal{O}_v$ and residue field $\kappa(v)$, the localization sequence of Quillen K-theory for the closed–open complement $\mathrm{Spec}\,\kappa(v)\to\mathrm{Spec}\,\mathcal{O}_v\leftarrow\mathrm{Spec}\,F$ produces a boundary map $\partial_v\colon K_2(F)\to\kappa(v)^{\times}$ given by the explicit tame symbol formula $\partial_v\{f,g\}=(-1)^{v(f)v(g)}f^{v(g)}/g^{v(f)}\bmod\mathfrak{m}_v$. Under the Teichmüller lift - the canonical multiplicative section carrying the roots of unity of the residue field into $\mathcal{O}_v^{\times}$ - this prime-to-p output agrees exactly with the prime-to-p part of the Hilbert symbol; the wild part is invisible to the localization sequence of the regular integral model $\mathcal{O}_v$.

Gillet showed that Weil reciprocity - for $f,g$ in the function field of a smooth proper curve $X$ over a field $k$, the product over closed points $x$ of the normed tame symbols $N_{\kappa(x)/k}(\partial_x\{f,g\})$ equals $1$ - follows from the K-theory localization sequence: the composite $K_2(k(X))\to\bigoplus_x K_1(\kappa(x))\to K_1(k)$ is null because it factors through $K_1(X)$ (the proof is reproduced in Weibel's K-book as the Weil Reciprocity Formula and in Braunling's paper). For $k=\mathbb{F}_q$ this yields Hilbert reciprocity for global function fields, since there the roots of unity are prime to the characteristic and the Hilbert symbol coincides with the tame symbol. Hilbert reciprocity for a number field $F$, the counterpart under the function field/number field dictionary, states that $\prod_v h_v(\alpha,\beta)^{m_v/m}=1$ for $\alpha,\beta\in F^{\times}$ ($m=\#\mu(F)$, over noncomplex places); Moore strengthened this to exactness of the Moore sequence $K_2(F)\to\bigoplus_{v\text{ noncomplex}}\mu(F_v)\to\mu(F)\to 0$.

Transposing Gillet's argument to $X=\mathrm{Spec}\,\mathcal{O}_F$ fails for identified reasons. The tame boundary only reaches the prime-to-p part of each $\mu(F_v)$, while Hilbert reciprocity crucially hinges on wild ramification; at $\mathbb{Q}_2$ the tame symbol $K_2(\mathbb{Q}_2)\to\mathbb{F}_2^{\times}$ is even trivial. Moreover Soulé's theorem on the K-theory of rings of integers of number fields ($K_n(\mathcal{O}_F)\simeq K_n(F)$ for odd $n\ge 3$, with the even-degree boundary maps being isomorphisms) forces the naive analogue of Gillet's diagram to degenerate, so the localization sequence of the regular model teaches nothing new. An order in $F$ is a subring $R\subseteq\mathcal{O}_F$, finitely generated as an abelian group, with fraction field $F$; the ring of integers is the maximal order. Braunling's idea is to shrink to a deliberately singular order near the ramification locus ('pinching singularities'): K-theory with support at the pinched ideal - the relative groups $K_{I,n}(R)$, defined as the K-theory of perfect complexes of $R$-modules which become acyclic after pulling them back to the open $\mathrm{Spec}\,R-\mathrm{Spec}\,R/I$, in place of the K-theory of the residue ring $R/I$ - then fattens up, and at odd $p$ the resulting boundary map is exactly the full local Hilbert symbol, with $K_{\tilde{m},1}(R)\simeq\mu(F_v)$. Run adelically this identifies the localization sequence of a global optimal order with Moore's sequence up to 2-primary torsion. What is missing at places above $2$ and at real places is the subject of the present problem; $SK_1(R)$ denotes the kernel of $K_1(R)\to K_1(F)$ for an order $R$ in $F$.

Problem Statement

Let $F$ be a number field. Realize the full local Hilbert symbol as the boundary map of a K-theory localization sequence such that: (1) at every noncomplex place $v$ of $F$, including the finite places above the prime $2$ and the real (archimedean) places, the boundary map on $K_2$ arising from that localization sequence is the full local Hilbert symbol $h_v\colon K_2(F_v)\to\mu(F_v)$, including its p-primary wild part at the finite places (the part invisible to the tame symbol; over $\mathbb{Q}_2$ the tame symbol is trivial) and its sign part at the real places; and (2) globally, the resulting localization sequence is identified with Moore's exact sequence $K_2(F)\to\bigoplus_{v\text{ noncomplex}}\mu(F_v)\to\mu(F)\to 0$, so that Hilbert reciprocity $\prod_v h_v(\alpha,\beta)^{m_v/m}=1$ becomes the $d^2=0$ property of a K-theory localization sequence, without inverting $2$. Also identify $SK_1(R)\simeq\mu(F)$ for the resulting global order without circular recourse to global class field theory. The current frontier is precise: Braunling (2023) achieves (1) and (2) at all finite places above odd primes and up to 2-primary torsion; what is asked for is the completion at the places above $2$ and at the real places.

References

1. Oliver Braunling, 'Hilbert reciprocity using K-theory localization', Pure and Applied Mathematics Quarterly 19 (2023), no. 2, 409-450. DOI: 10.4310/pamq.2023.v19.n2.a1; arXiv:2111.11580; https://arxiv.org/abs/2111.11580
2. Evgeny Musicantov, Alexander Yom Din, 'Reciprocity laws and K-theory', Annals of K-Theory 2 (2017), no. 1, 27-46. DOI: 10.2140/akt.2017.2.27; arXiv:1410.5391; https://arxiv.org/abs/1410.5391
3. Dustin Clausen, 'p-adic J-homomorphisms and a product formula', arXiv:1110.5851 (2011, revised 2012); https://arxiv.org/abs/1110.5851
4. Dustin Clausen, 'A K-theoretic approach to Artin maps', arXiv:1703.07842 (2017); https://arxiv.org/abs/1703.07842
5. Christophe Soulé, 'K-théorie des anneaux d'entiers de corps de nombres et cohomologie étale', Inventiones Mathematicae 55 (1979), no. 3, 251-295. DOI: 10.1007/BF01406843; https://doi.org/10.1007/BF01406843
6. Charles Weibel, 'The K-book: An Introduction to Algebraic K-theory', Graduate Studies in Mathematics 145, American Mathematical Society, 2013. DOI: 10.1090/gsm/145; ISBN 978-0-8218-9134-2; https://doi.org/10.1090/gsm/145
7. Henri Gillet, 'Riemann-Roch theorems for higher algebraic K-theory', Advances in Mathematics 40 (1981), 203-289. DOI: 10.1016/S0001-8708(81)80006-0; https://doi.org/10.1016/S0001-8708(81)80006-0
8. Peter Arndt, Oliver Braunling, 'On the automorphic side of the K-theoretic Artin symbol', Selecta Mathematica (New Series) 25 (2019), no. 3, Paper No. 38. DOI: 10.1007/s00029-019-0485-8; https://doi.org/10.1007/s00029-019-0485-8
9. Oliver Braunling, Ruben Henrard, Adam-Christiaan van Roosmalen, 'A non-commutative analogue of Clausen's view on the idèle class group', arXiv:2109.04331 (2021); https://arxiv.org/abs/2109.04331
10. Matteo Tamiozzo, 'The tame Hilbert symbol via K-theory and central extensions', Journal of Number Theory 242 (2023), 603-625. DOI: 10.1016/j.jnt.2022.05.003; arXiv:2010.07682; https://arxiv.org/abs/2010.07682
11. Jonathan Holland, 'Reciprocity and the Maslov Phase', arXiv:2604.25288 (2026); https://arxiv.org/abs/2604.25288