The tame Hilbert symbol via K-theory and central extensions

TL;DR

This paper develops a K-theoretic framework for the tame Hilbert symbol over local fields and constructs related central extensions of general linear groups, linking residue symbols to group extensions.

Contribution

It introduces a novel K-theoretic enhancement of the n-th power residue symbol and constructs explicit central extensions of GL_m(K) related to these symbols.

Findings

Constructed a K-theoretic enhancement of the n-th power residue symbol.

Built central extensions of GL_m(K) by μ_n linked to residue symbols.

Expressed residue symbols using extensions derived from the case m=1.

Abstract

Let $K$ be a mixed characteristic local field whose residue field has cardinality $q$, and let $n$ be an integer dividing $q-1$. In the first part of this document we construct a $K$-theoretic enhancement of the $n$-th power residue symbol $K^\times \times K^\times \rightarrow \mu_n$. In the second part we construct central extensions of $GL_m(K)$ by $\mu_n$ and we express the $n$-th power residue symbol in terms of a symbol defined using the extension obtained for $m=1$. Our constructions both rely on the study of finite free pointed $\mu_n$-sets.