Reciprocity and orthogonality

TL;DR

This paper investigates the structure of a natural filtration and a bilinear pairing on certain multiplicative groups of local fields, determining their orthogonal filtrations and extending results to characteristic p fields.

Contribution

It explicitly determines the orthogonal filtration associated with the reciprocity pairing on local fields and extends the analysis to fields of characteristic p.

Findings

Explicit description of the orthogonal filtration for the pairing

Extension of results to characteristic p fields

Insight into the structure of multiplicative groups in local fields

Abstract

Let $p$ be a prime and let $K$ be a finite extension of the field ${\bf Q}_p$ of $p$-adic numbers such that the group ${}_pK^\times$ has order $p$. The ${\bf F}_p$-space $K^\times\!/K^{\times p}$ carries a natural filtration coming from the valuation on $K$, and a natural bilinear pairing coming from the reciprocity isomorphism for the exponent~$p$. We determine the orthogonal filtration for this pairing. We also prove the analogous result for $p$-fields of characteristic $p$.