Cohomology of ad\'{e}le class groups of algebraic tori

TL;DR

This paper computes the Herbrand quotient of the adèle class group of an algebraic torus over a global field, extending classical inequalities in global class field theory.

Contribution

It provides a new explicit calculation of the Herbrand quotient for tori, generalizing key inequalities in global class field theory.

Findings

Herbrand quotient of the adèle class group is determined

Extension of the first inequality of global class field theory

Results apply to tori split over finite cyclic extensions

Abstract

Let $T$ be an algebraic torus defined over a global field $K$ and split over a finite cyclic extension. In this paper, we determine the Herbrand quotient of the ad\'{e}le class group of $T$. Our result can be seen as an extension of the so-called first inequality of global class field theory.