Let $k$ be a field, $T$ be an algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and ${\rm Pic}\,\overline{X}$ be the Picard group of $\overline{X}=X\times_k\overline{k}$. Hoshi, Kanai and Yamasaki [HKY22] determined $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with $[K:k]=n\leq 15$ and $n\neq 12$. In this paper, we determine $64$ cases with $H^1(k,{\rm Pic}\, \overline{X})\neq 0$ and give a necessary and sufficient condition for the Hasse norm principle for $K/k$ where $[K:k]=12$.
The aim of this note is to revisit the question of local-global principles for embeddings of etale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and sufficient condition is given in a joint paper with Tingyu Lee and Raman Parimala. In the present paper, we give a simpler description of the obstruction group. It is also shown that if the etale algebra is a product of pairwise independent field extensions, then the Hasse principle holds.
The aim of this paper is to revisit the question of local-global principles for embeddings of étale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and sufficient condition is given in [BLP 18]. In the present paper, we give a simpler description of the obstruction group. It is also shown that if the etale algebra is a product of pairwise linearly disjoint field extensions, then the Hasse principle holds, and that if an embedding exists after an odd degree extension, then it also exists over the global field itself. …
We show that certain Tate--Shafarevich groups are unramified which enables us to give an obstruction to the Hasse principle for torsors under tori over p-adic function fields.
Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global field $k$, Voskresenskii proved that there exists an exact sequence $0\to A(T)\to H^1(k,{\rm Pic}\,\overline{X})^\vee\to Sha(T)\to 0$ where $A(T)$ is the kernel of the weak approximation of $T$, $Sha(T)$ is the Shafarevich-Tate group of $T$, $X$ is a smooth $k$-compactification of $T$, $\overline{X}=X\times_k\overline{k}$, ${\rm Pic}\,\overline{X}$ is the Picard group of $\overline{X}$ and $\vee$ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus $T=R^{(1)}_{K/k}(G_m)$ o…
Let $K$ be a global field of positive characteristic. We prove that the Brauer-Manin obstructions to the Hasse principle, to weak approximation and to strong approximation are the only ones for homogeneous spaces of reductive groups with reductive stabilizers. The methods involve abelianization techniques and arithmetic duality theorems for complexes of tori over K.
We classify stably/retract rational norm one tori in dimension $p-1$ where $p$ is a prime number and in dimension up to ten with some minor exceptions.
We classify stably/retract rational norm one tori in dimension $n-1$ for $n=2^e$ $(e\geq 1)$ is a power of $2$ and $n=12, 14, 15$. Retract non-rationality of norm one tori for primitive $G\leq S_{2p}$ where $p$ is a prime number and for the five Mathieu groups $M_n\leq S_n$ $(n=11,12,22,23,24)$ is also given.
In this paper we introduce a method to obtain algebraic information using arithmetic one in the study of tori and their principal homogeneous spaces. In particular, using some results of the authors with Tingyu Lee, we determine the unramified Brauer groups of some norm one tori, and their torsors.
