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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 $\overline{X}=X\times_k\overline{k}$ where $\overline{k}$ is a fixed separable closure of $k$. Hoshi, Kanai and Yamasaki [HKY22], [HKY23] determined $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with $[K:k]\leq 15$. In this paper, we treat the case where $[K:k]=16$. Among $1954$ transitive subgroups $G=16Tm\leq S_{16}$ $(1\leq m\leq 1954)$ up to conjugacy, we determine $1101$ (resp. $774$, $31$, $37$, $1$, $1$, $9$) cases with $H^1(k,{\rm Pic}\, \overline{X})=0$ (resp. $Z/2Z$, $(Z/2Z)^{\oplus 2}$, $(Z/2Z)^{\oplus 3}$, $(Z/2Z)^{\oplus 4}$, $(Z/2Z)^{\oplus 6}$, $Z/4Z$) where $G$ is the Galois group of the Galois closure $L/k$ of $K/k$. We see that $H^1(k,{\rm Pic}\, \overline{X})=0$ implies that the Hasse norm principle holds for $K/k$. In particular, among $22$ primitive $G=16Tm$ cases, i.e. $H\leq G=16Tm$ is maximal with $[G:H]=16$, we determine exactly $6$ cases $(m=178, 708, 1080, 1329, 1654, 1753)$ with $H^1(k,{\rm Pic}\, \overline{X})\neq 0$ $($$(Z/2Z)^{\oplus 2}$, $Z/2Z$, $(Z/2Z)^{\oplus 2}$, $Z/2Z$, $Z/2Z$, $Z/2Z$). Moreover, we give a necessary and sufficient condition for the Hasse norm principle for $K/k$ with $[K:k]=16$ for $22$ primitive $G=16Tm$ cases. As a consequence of the $22$ primitive $G$ cases, we get the Tamagawa number $\tau(T)=1$, $1/2$, $1/4$ of $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ over a number field $k$ via Ono's formula $\tau(T)=1/|Sha(T)|$ where $Sha(T)$ is the Shafarevich-Tate group of $T$., Comment: To appear in J. Algebra, 75 pages, modified Section 5 and added Norm1ToriHNP for GAP 4 ver.2024.04.03 to references which is available from KURENAI (Kyoto University Research Information Repository) https://doi.org/10.57723/289563

In this paper, we introduce a kind of decomposition of a finite group called a uniform group factorization, as a generalization of exact factorizations of a finite group. A group $G$ is said to admit a uniform group factorization if there exist subgroups $H_1, H_2, \ldots, H_k$ such that $G = H_1 H_2 \cdots H_k$ and the number of ways to represent any element $g \in G$ as $g = h_1 h_2 \cdots h_k$ ($h_i \in H_i$) does not depend on the choice of $g$. Moreover, a uniform group factorization consisting of cyclic subgroups is called a uniform cyclic group factorization. First, we show that any finite solvable group admits a uniform cyclic group factorization. Second, we show that whether all finite groups admit uniform cyclic group factorizations or not is equivalent to whether all finite simple groups admit uniform group factorizations or not. Lastly, we give some concrete examples of such factorizations., Comment: 10 pages. To appear in Communications in Algebra

We give a necessary and sufficient condition for the Hasse norm principle for field extensions $K/k$ when the Galois groups ${\rm Gal}(L/k)$ of the Galois closure $L/k$ of $K/k$ are isomorphic to the Mathieu group $M_{11}$ of degree $11$ of order $7920$ or the Janko group $J_1$ of order $175560$ by determining $H^1(k,{\rm Pic}\, \overline{X})=0$ or $\mathbb{Z}/2\mathbb{Z}$ for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ with a smooth $k$-compactification $X$ and $\overline{X}=X\times_k\overline{k}$. The result gives a first step towards understanding the all pictures of the Hasse norm principle for the $26$ sporadic simple groups., Comment: 18 pages, added Norm1ToriHNP for GAP 4 ver.2024.04.03 to references which is available from KURENAI (Kyoto University Research Information Repository) http://hdl.handle.net/2433/289563

Let $e \geq 2$ be an integer, $p^r$ be a prime power with $p^r \equiv 1\ ({\rm mod}\ e)$ and $\eta_r(i)$ be Gaussian periods of degree $e$ for ${\mathbb F}_{p^r}$. By the dual form of Davenport and Hasse's lifting theorem on Gauss sums, we establish lifts of the multiplication matrices of the Gaussian periods $\eta_r(0),\ldots,\eta_r(e-1)$ which are defined by F. Thaine. We also give some examples of the explicit lifts for prime degree $e$ with $3\leq e\leq 23$ which also illustrate relations among lifts of Jacobi sums, Gaussian periods and multiplication matrices of Gaussian periods., Comment: To appear in Finite Fields Appl., 28 pages

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 $[K:k]=12$., Comment: To appear in J. Number Theory, 148 pages

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 $T=R^{(1)}_{K/k}(G_m)$ of $K/k$, $Sha(T)=0$ if and only if the Hasse norm principle holds for $K/k$. First, we determine $H^1(k,{\rm Pic}\, \overline{X})$ for algebraic $k$-tori $T$ up to dimension $5$. Second, we determine $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ with $[K:k]=n\leq 15$ and $n\neq 12$. We also show that $H^1(k,{\rm Pic}\, \overline{X})=0$ for $T=R^{(1)}_{K/k}(G_m)$ when the Galois group of the Galois closure of $K/k$ is the Mathieu group $M_n\leq S_n$ with $n=11,12,22,23,24$. Third, we give a necessary and sufficient condition for the Hasse norm principle for $K/k$ with $[K:k]=n\leq 15$ and $n\neq 12$. As applications of the results, we get the group $T(k)/R$ of $R$-equivalence classes over a local field $k$ via Colliot-Th\'{e}l\`{e}ne and Sansuc's formula and the Tamagawa number $\tau(T)$ over a number field $k$ via Ono's formula $\tau(T)=|H^1(k,\widehat{T})|/|Sha(T)|$., Comment: To appear in Math. Comp., 98 pages