Asian Journal of Mathematics

Volume 21 (2017)

Number 2

Function fields of algebraic tori revisited

Pages: 197 – 224



Shizuo Endo (Department of Mathematics, Tokyo Metropolitan University, Tokyo, Japan)

Ming-Chang Kang (Department of Mathematics, National Taiwan University, Taipei, Taiwan)


Let $K/k$ be a finite Galois extension and $\pi = \mathrm{Gal}(K/k)$. An algebraic torus $T$ defined over $k$ is called a $\pi$-torus if $T \times {}_{\mathrm{Spec}(k)} \: \mathrm{Spec}(K) \simeq \mathbb{G}^n_{m,K}$ for some integer $n$. The set of all algebraic $\pi$-tori defined over $k$ under the stably birational equivalence forms a semigroup, denoted by $T(\pi)$. We will give a complete proof of the following theorem due to Endo and Miyata [EM3]. Theorem. Let $\pi$ be a finite group. Then $T(\pi) \simeq C(\Omega_{\mathbb{Z} \pi})$ where $(\Omega_{\mathbb{Z} \pi}$ is a maximal $\mathbb{Z}$-order in $\mathbb{Q} \pi$ containing $\mathbb{Z} \pi$ and $C (\Omega_{\mathbb{Z}_\pi})$ is the locally free class group of $\Omega_{\mathbb{Z} \pi}$, provided that $\pi$ is isomorphic to one of the following four types of groups: $C_n$ ($n$ is any positive integer), $D_m$ ($m$ is any odd integer $\geq 3$), $C_{q^f} \times D_m$ ($m$ is any odd integer $\geq 3$, $q$ is an odd prime number not dividing $m$, $f \geq 1$, and $(\mathbb{Z} / q^f \; \mathbb{Z})^{\times} = \langle \bar{p} \rangle$ for any prime divisor $p$ of $m$), $Q_{4m}$ ($m$ is any odd integer $\geq 3, p \equiv 3 (\mathrm{mod} \: 4)$ for any prime divisor $p$ of $m$).


algebraic torus, rationality problem, locally free class groups, class numbers, maximal orders, twisted group rings

2010 Mathematics Subject Classification

11R29, 11R33, 14E08, 20C10

Full Text (PDF format)

Received 23 January 2015

Published 5 July 2017