Algebraic tori revisited

Let $K/k$ be a finite Galois extension and \pi = \fn{Gal}(K/k). An algebraic torus $T$ defined over $k$ is called a $\pi$-torus if T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n for some integer $n$. The set of all algebraic $\pi$-tori defined over $k$ under the stably isomorphism form a semigroup, denoted by $T(\pi)$. We will give a complete proof of the following theorem due to Endo and Miyata \cite{EM5}. Theorem. Let $\pi$ be a finite group. Then $T(\pi)\simeq C(\Omega_{\bm{Z}\pi})$ where $\Omega_{\bm{Z}\pi}$ is a maximal $\bm{Z}$-order in $\bm{Q}\pi$ containing $\bm{Z}\pi$ and $C(\Omega_{\bm{Z}\pi})$ is the locally free class group of $\Omega_{\bm{Z}\pi}$, provided that $\pi$ is isomorphic to the following four types of groups : $C_n$ ($n$ is any positive integer), $D_m$ ($m$ is any odd integer $\ge 3$), $C_{q^f}\times D_m$ ($m$ is any odd integer $\ge 3$, $q$ is an odd prime number not dividing $m$, $f\ge 1$, and $(\bm{Z}/q^f\bm{Z})^{\times}=\langle \bar{p}\rangle$ for any prime divisor $p$ of $m$), $Q_{4m}$ ($m$ is any odd integer $\ge 3$, $p\equiv 3 \pmod{4}$ for any prime divisor $p$ of $m$).Comment: To appear in Asian J. Math. ; the title is change

{Twisted Poincar\'e Series and Geometric Factorization of Affine Weyl Groups

We study the relation between the alternating product of twisted Poincar'e series of parabolic subgroups of affine Weyl groups and hyperbolic stabilizers of geodesic tubes. Moreover, from such relation, we find a length-preserving factorization of affine Weyl groups of type $\tilde{A}_n$ and $\tilde{C}_n$

Retract Rational Fields

Let $k$ be an infinite field. The notion of retract $k$-rationality was introduced by Saltman in the study of Noether's problem and other rationality problems. We will investigate the retract rationality of a field in this paper. Theorem 1. Let $k\subset K\subset L$ be fields. If $K$ is retract $k$-rational and $L$ is retract $K$-rational, then $L$ is retract $k$-rational. Theorem 2. For any finite group $G$ containing an abelian normal subgroup $H$ such that $G/H$ is a cyclic group, for any complex representation $G \to GL(V)$, the fixed field $\bm{C}(V)^G$ is retract $\bm{C}$-rational. Theorem 3. If $G$ is a finite group, then all the Sylow subgroups of $G$ are cyclic if and only if $\bm{C}_{\alpha}(M)^G$ is retract $\bm{C}$-rational for all $G$-lattices $M$, for all short exact sequences $\alpha : 0 \to \bm{C}^{\times} \to M_{\alpha} \to M \to 0$. Because the unramified Brauer group of a retract $\bm{C}$-rational field is trivial, Theorem 2 and Theorem 3 generalize previous results of Bogomolov and Barge respectively (see Theorem \ref{t5.9} and Theorem \ref{t6.1}).Comment: Several typos in the previous version were correcte