Local–global principle for reduced norms over function fields of $p$ -adic curves
Local–global principle for reduced norms over function fields of $p$ -adic curves
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DOI：
10.1112/s0010437x17007618
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发表时间：
2016-05
影响因子：
1.8
通讯作者：
R. Parimala;R. Preeti;V. Suresh
中科院分区：
文献类型：
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作者：
R. Parimala;R. Preeti;V. Suresh
设K$为（非阿基米德）局部域，设F$为K$上曲线的函数域。设$D$是周期$n$和$\unicode[STIX]{x1D706}\in F^{\ast}$的$F$上的一个中心简单代数。我们证明了如果$n$是$K$和$D\cdot (\unicode[STIX]{x1D706})=0$在$H^{3}(F,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$中的残差域特征的协素数，则$\unicode[STIX]{x1D706}$是$D$的约简范数。这导致了群$\operatorname{SL}_{1}(D)$的Hasse原则，即F^{\ast}$中的元素$\unicode[STIX]{x1D706}\是$D$的约简范数，当且仅当它是$F$的所有离散赋值处的局部约简范数。
Let $K$ be a (non-archimedean) local field and let $F$ be the function field of a curve over $K$ . Let $D$ be a central simple algebra over $F$ of period $n$ and $\unicode[STIX]{x1D706}\in F^{\ast }$ . We show that if $n$ is coprime to the characteristic of the residue field of $K$ and $D\cdot (\unicode[STIX]{x1D706})=0$ in $H^{3}(F,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$ , then $\unicode[STIX]{x1D706}$ is a reduced norm from $D$ . This leads to a Hasse principle for the group $\operatorname{SL}_{1}(D)$ , namely, an element $\unicode[STIX]{x1D706}\in F^{\ast }$ is a reduced norm from $D$ if and only if it is a reduced norm locally at all discrete valuations of $F$ .