Norm one tori and Hasse norm principle, II: Degree 12 case
Norm one tori and Hasse norm principle, II: Degree 12 case
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范数一环面和哈斯范数原理，II：12 阶情况
DOI：
10.1016/j.jnt.2022.09.006
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发表时间：
2023
影响因子：
0.7
通讯作者：
A.Yamasaki
中科院分区：
文献类型：
--
作者：
A. Hoshi;K. Kanai;A.Yamasaki
设k是一个域，T是一个代数k-环面，X是T的光滑k-紧化，Pic X ∈是X ∈ = X× kk ∈的Picard群. Hoshi，Kanai和Yamasaki [HKY 22]确定了范数为1的环面T= RK/k（1）（Gm）的H1（k，Pic X），并给出了数域扩张K/k满足[K：k]= n≤ 15和n ≤ 12的Hasse范数原理的一个充要条件.本文确定了64种H1（k，Pic X）≥ 0的情形，并给出了K/k满足[K：k]= 12时Hasse范数原理成立的充要条件.
Let k be a field, T be an algebraic k-torus, X be a smooth k-compactification of T and Pic X‾ be the Picard group of X‾= X× k k‾. Hoshi, Kanai and Yamasaki [HKY22] determined H 1 (k, Pic X‾) for norm one tori T= R K/k (1)(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≤ 15 and n≠ 12. In this paper, we determine 64 cases with H 1 (k, Pic X‾)≠ 0 and give a necessary and sufficient condition for the Hasse norm principle for K/k with [K: k]= 12.