Hilbert reciprocity using K-theory localization

TL;DR

This paper develops a novel approach to understanding Hilbert reciprocity in number fields by modifying K-theory boundary maps to include wild ramification effects, except at p=2.

Contribution

It introduces a method to incorporate wild ramification into K-theory boundary maps, extending the scope of Hilbert reciprocity beyond tame cases.

Findings

Successfully visualizes wild symbols as boundary maps in K-theory

Extends Hilbert reciprocity to number fields with wild ramification (except p=2)

Provides a new perspective on ramification effects in algebraic K-theory

Abstract

Usually the boundary map in K-theory localization only gives the tame symbol at $K_{2}$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at p=2. Our idea is to pinch singularities near the ramification locus. This fattens up K-theory and makes the wild symbol visible as a boundary map.