Norm one tori and Hasse norm principle
Norm one tori and Hasse norm principle
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范数一托里和哈斯范数原理
DOI：
10.1090/mcom/3735
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发表时间：
2022
影响因子：
2
通讯作者：
A.Yamasaki
中科院分区：
文献类型：
--
作者：
A. Hoshi;K. Kanai;A.Yamasaki
设是一个域，是一个代数环面。1969年，Voskresenskiy证明了在整体域上存在正合序列0\to A（T）\to H^ 1（k，\operatorname {Pic}\overline {X}）^\vee\to\Sha（T）\to 0$其中是的弱逼近的核，$\Sha（T）$是的Shafarevich-Tate群，是的光滑紧化，是的Picard群，代表Pontryagin对偶。另一方面，在1963年，小野证明了，对于一个torusof的范数，$\Sha（T）= 0$当且仅当Hasse范数原理成立。首先，我们确定代数拓扑的维数。其次，我们确定范数为1的toriwithand。我们还证明了当的Galois闭包的Galois群是Mathieu群时.第三，给出了带和的Hasse范数原理成立的一个充要条件.作为这些结果的应用，我们利用Colliot-Thélène和Santiago公式得到了局部域上的群等价类，利用Ono公式得到了数域上的Tamagawa数|H^ 1（k，\widehat {T}）|/|\沙（T）| $.引用
Letbe a field andbe an algebraic-torus. In 1969, over a global field, Voskresenskiǐ proved that there exists an exact sequence $0\to A (T)\to H^ 1 (k,\operatorname {Pic}\overline {X})^\vee\to\Sha (T)\to 0$ whereis the kernel of the weak approximation of, $\Sha (T) $ is the Shafarevich-Tate group of,is a smooth-compactification of,,is the Picard group ofandstands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torusof, $\Sha (T)= 0$ if and only if the Hasse norm principle holds for. First, we determinefor algebraic-toriup to dimension. Second, we determinefor norm one toriwithand. We also show thatforwhen the Galois group of the Galois closure ofis the Mathieu groupwith. Third, we give a necessary and sufficient condition for the Hasse norm principle forwithand. As applications of the results, we get the groupof-equivalence classes over a local fieldvia Colliot-Thélène and Sansuc’s formula and the Tamagawa numberover a number fieldvia Ono’s formula $\tau (T)=| H^ 1 (k,\widehat {T})|/|\Sha (T)| $. References