Rationality problem for norm one tori of tensor products of étale algebras and Hasse norm principle

Abstract

Let $k$ be a field. Let $A={\prod }_{i=1}^r{K}_i$ and $B={\prod }_{j=1}^s{E}_j$ be étale $k$-algebras where $K_i$ and $E_j$ are finite separable field extensions of $k$ with $[K_i:k]=m_i$ and $[E_j:k]=n_j$. Let ${\mathcal{T}}_A=R_{A/k}^{(1)}({\mathbb{G}}_m)$ be the norm one torus of the étale $k$-algebra $A$. We prove that if $\gcd (m_i,n_j\mid 1\le i\le r,1\le j\le s)=1$ and ${\mathcal{T}}_A$ and ${\mathcal{T}}_B$ are stably $($resp. retract$)$ $k$-rational, then the algebraic $k$-torus ${\mathcal{T}}_A\otimes {\mathcal{T}}_B$ and the norm one torus ${\mathcal{T}}_{A\otimes B}$ are stably $($resp. retract$)$ $k$-rational. In particular, if $k$ is a global field, then the Hasse norm principle holds for $(A\otimes B)/k$. We introduce a new invariant of $G$-lattices, the permutation order, whose triviality is equivalent to invertibility, and use it to study the rationality of tensor products $T_1\otimes T_2$ of algebraic $k$-tori. As an application, we obtain large families of field extensions $K/k$ for which the Hasse norm principle holds.

38 pages, abstract, Proposition 6.5 and the proof of Proposition 6.6 are modified