论文标题
Norm One Tori and Hasse Norm Arimiple，II：学位$ 12 $案例
Norm one tori and Hasse norm principle, II: Degree $12$ case
论文作者
论文摘要
让$ k $成为一个字段，$ t $为代数$ k $ -torus，$ x $是$ t $的平滑$ k $ -compactification，$ t $和$ {\ rm pic} \，\ overline {x} $是$ \ edline {x}的picard {x} $的picard组。 hoshi，kanai和yamasaki [hky22]确定$ h^1（k，{\ rm pic} \，\ edlline {x}）$ for norm One tori $ t = r^r^{（1）_ {k/k/k/k/k}（g_m）$，并给出了足够的条件，并提供了k. $ k $ k $ k $ k的/ $ [k：k] = n \ leq 15 $和$ n \ neq 12 $。在本文中，我们确定$ H^1（k，{\ rm pic} \，\ overline {x}）\ neq 0 $的$ 64 $ case，并为$ k/k $ where $ [k：k：k] = 12 $的hasse norm instiriped of Husse Normiquie of neq 0 $ uneq 0 $。
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 $[K:k]=12$.
