Reciprocity and orthogonality

Abstract

Let $p$ be a prime and let $K$ be a finite extension of the field ${\mathbf{Q}}_p$ of $p$-adic numbers such that the group ${}_p{K}^{\times }$ has order $p$. The ${\mathbf{F}}_p$-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 $p$.

7 pages

References in corpus (2)

• Further remarks on local discriminants
• Local discriminants, kummerian extensions, and elliptic curves