Class numbers of multinorm-one tori

TL;DR

This paper derives a general formula for the class number of multinorm-one tori over global fields, extending Ono's invariants and generalizing classical genus theory results.

Contribution

It introduces a comprehensive formula for class numbers of multinorm-one tori, generalizing Ono's invariants and classical genus theory to arbitrary étale algebras over global fields.

Findings

Derived a formula for class numbers of multinorm-one tori.

Extended Ono's invariants to arbitrary S-ideal class numbers.

Generalized classical results of genus theory.

Abstract

We present a formula for the class number of a multinorm one torus $T_{L/k}$ associated to any \'etale algebra $L$ over a global field $k$. This is deduced from a formula for analogues of invariants introduced by T.~Ono, which are interpreted as a generalization of Gauss genus theory. This paper includes the variants of Ono's invariant for arbitrary $S$-ideal class numbers and the narrow version, generalizing results of Katayama, Morishita, Sasaki and Ono.