The Discriminant of an Algebraic Torus

TL;DR

This paper revisits and refines the class number formula for algebraic tori over global fields, providing a new local-global perspective on the quasi-discriminant using cocharacters.

Contribution

It offers a new local-global presentation of the quasi-discriminant of a torus, enhancing the understanding of its role in class number formulas.

Findings

Established a local-global description of the quasi-discriminant

Connected the quasi-discriminant to cocharacters of the torus

Provided a more natural definition of the invariant

Abstract

For a torus T defined over a global field K, we revisit an analytic class number formula obtained by Shyr in the 1970's as a generalization of Dirichlet's class number formula. We prove a local-global presentation of the quasi-discriminant of T, which enters into this formula, in terms of cocharacters of T. This presentation can serve as a more natural definition of this invariant.