半整体域上Hermite空间的局部-整体原理
Local-global principles for hermitian spaces over semi-global fields
摘要：
设$K$是一个具有剩余域$k$的完备离散值域，$F$是$K$上曲线的函数域.设$A \in {}_2Br(F)$是一个中心单代数，其对合$σ$为任意类型，且有$F_0 =F^σ$ .设$h$是$(A, σ)$和$G = SU(A, σ, h)$上的厄米特空间，如果$σ$是第一类的，而$G = U(A, σ, h)$是第二类的。假设$\text{char}(k) \neq 2$和ind $(A)\leq 4$ .然后证明了$F_0$上$G$下的射影齐性空间对于$F$的离散赋值的有理点满足局部-整体原理。
英文摘要：
Let $K$ be a complete discrete valued field with residue field $k$ and $F$ the function field of a curve over $K$. Let $A \in {}_2Br(F)$ be a central simple algebra with an involution $σ$ of any kind and $F_0 =F^σ$. Let $h$ be an hermitian space over $(A, σ)$ and $G = SU(A, σ, h)$ if $σ$ is of first kind and $G = U(A, σ, h)$ if $σ$ is of second kind. Suppose that $\text{char}(k) \neq 2$ and ind$(A)\leq 4$. Then we prove that projective homogeneous spaces under $G$ over $F_0$ satisfy a local-global principle for rational points with respect to discrete valuations of $F$.
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