Hilbert reciprocity using $K$-theory localization

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.

Keywords

Hilbert reciprocity law, Moore sequence, localization sequence, Hilbert symbol, tame symbol

2010 Mathematics Subject Classification

Primary 11A15, 11S70. Secondary 19C20.

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The author was supported by the EPSRC Programme Grant EP/M024830/1 “Symmetries and correspondences: intra-disciplinary developments and applications”.

Received 5 August 2022

Received revised 2 December 2022

Accepted 2 February 2023

Published 7 April 2023