On an explicit reciprocity law in local class field theory via $(\varphi, \Gamma)$-modules

TL;DR

This paper derives an explicit formula for the Hilbert symbol over certain local fields using the theory of $(, \Gamma)$-modules, advancing explicit reciprocity laws in local class field theory.

Contribution

It provides a new explicit formula for the Hilbert symbol over unramified extensions of $Q_2$ utilizing $(, \Gamma)$-modules, connecting local class field theory with modern module theory.

Findings

Explicit formula for the $oldsymbol{rac{2^n}{K_n}}$-valued Hilbert symbol.

Application of $(, \\Gamma)$-modules to explicit reciprocity laws.

Enhanced understanding of local class field theory in the 2-adic setting.

Abstract

Let $K$ be an unramified extension of $\mathbb{Q}_2$ and $\mu_{2^n}$ the group of $2^n$-th root of unity for a fixed integer $n\geqslant 2$. In this paper, we give an explicit formula for the $\mu_{2^n}$-valued Hilbert symbol over $K_n := K(\mu_{2^n})$ using the theory of $(\varphi, \Gamma)$-modules.