Function fields of algebraic tori revisited

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 $m$).

Keywords

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

2010 Mathematics Subject Classification

11R29, 11R33, 14E08, 20C10

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Received 23 January 2015

Published 5 July 2017