Twisted Class Field Theory

TL;DR

This paper explores an alternative $oldsymbol{ ext{ exttwosuperior}}$-extension of $oldsymbol{ ext{ extQp}}$, demonstrating that the axioms of Neukirch's class field theory hold and that the reciprocity maps coincide for both extensions.

Contribution

It proves that Neukirch's axioms are valid when using the alternative $oldsymbol{ ext{ extQp}}$ extension and shows the reciprocity maps are identical for both $oldsymbol{ ext{ extZhat}}$-extensions.

Findings

The axioms of class field theory hold for the alternative $ ext{ extQp}$ extension.

The reciprocity maps from the two $ ext{ extZhat}$-extensions are the same.

The result extends Neukirch's explicit class field theory to a new $ ext{ extQp}$ extension.

Abstract

Neukirch has developed explicit and axiomatic class field theory, which applies to both local and global fields. One of the key ingredients in his theory is a $\hat{\mathbb{Z}}$-extension of the base field, and in the case of $\mathbb{Q}_p$, he uses the maximal unramified extension. However $\mathbb{Q}_p$ has another $\hat{\mathbb{Z}}$-extension, which we shall denote by $\hat{\mathbb{Q}}_p$. Thus, it is natural to ask if we could verify all the axioms required by taking $\hat{\mathbb{Q}}_p$ as the central object instead. We prove this is possible and the two reciprocity maps induced from the two distinct $\hat{\mathbb{Z}}$-extensions are the same.