范数表示的二次多项式的万有环面和值
Universal torsors and values of quadratic polynomials represented by norms
摘要：
令$K/k$是数域的扩张，令$P(t)$是$k$上的二次多项式。设$X$是由$P(t) = N_{K/k}(\mathbf{z})$定义的仿射簇。本文研究了$X$的Hasse原理和弱逼近。对于$[K:k]=4$和$P(t)$在$k$上不可约且在$K$中分裂的情形，证明了Hasse原理和弱逼近.对于具有任意$K$的$k=\mathbb{Q}$，我们证明了对Hasse原理和弱逼近的Brauer-Manin阻碍是唯一的.对于$[K:k]=4$和$P(t)$在$k$上不可约，我们确定了$X$的光滑真模型的Brauer群.在它是非平凡的情况下，我们展示了一个反例弱近似。
英文摘要：
Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(\mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three cases. For $[K:k]=4$ and $P(t)$ irreducible over $k$ and split in $K$, we prove the Hasse principle and weak approximation. For $k=\mathbb{Q}$ with arbitrary $K$, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For $[K:k]=4$ and $P(t)$ irreducible over $k$, we determine the Brauer group of smooth proper models of $X$. In a case where it is non-trivial, we exhibit a counterexample to weak approximation.
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