Obtaining the Hilbert symbol from cup product on motivic cohomology

Question

Let F be a number field. Does the Hilbert symbols at the various places of F arise from the cup product on the motivic cohomology groups of Spec(F)? And if so, is it possible to interpret Moore's reciprocity sequence $$ K_2(F)\to \bigoplus_{v}\mu(F_v)\to\mu(F)\to0 $$ in this setting?

Answer 1

The Hilbert symbol is a Steinberg symbol, i.e., it can be interpreted as a morphism from $K_2(F)\to\mu_n(F)$ where $F$ is a local field with $n$-th roots of unity $\mu_n(F)$. It can be viewed as composition of three operations, two of which are related to cup products. The first step is to use the cup product for motivic cohomology (albeit in a rather trivial way): for two units $a,b\in F^\times\cong {\rm H}^1(F^\times,\mathbb{Z}(1))$ their cup product is the symbol $\{a,b\}\in {\rm H}^2(F^\times,\mathbb{Z}(2))\cong K^{\rm M}_2(F)$. The second step applies the norm residue isomorphism/Galois symbol/Chern class map (which arises from changing the topology from Nisnevich to étale): $K^{\rm M}_2(F)/\ell\cong {\rm H}^2_{\rm ét}(F,\mu_\ell^{\otimes 2})$. Finally, we need to get from the etale cohomology group to $\mu_\ell(F)$; and this is based on having an inverse of cup-product with the local invariant in the Brauer group.

This description of the Hilbert symbol is the one that gives rise to generalized/higher Hilbert symbols in the following papers of Banaszak and Kahn:

• G. Banaszak: "Generalization of the Moore exact sequence and the wild kernel for higher K-groups" Compositio Math 86 (1993), 281-305. (link to Numdam)

• B. Kahn: "Algebraic K-theory and twisted reciprocity laws" K-Theory 30 (2003), 211--240.

There are also generalizations of the Moore reciprocity sequence to higher K-groups in these papers. Looking at the constructions in these papers may help to better understand in what ways the classical Hilbert symbol and Moore reciprocity sequence fit into the motivic cohomology framework. For now, it seems to me that changing the topology from Nisnevich to etale (and using properties in etale cohomology like the Poitou-Tate sequence) is more fundamental than the use of the motivic cohomology cup product.