论文标题
半全球田地多型托里的本地全球原则
Local-global principles for multinorm tori over semi-global fields
论文作者
论文摘要
让$ k $成为一个完全离散价值的领域，而残留场$κ$。假设$κ$的同子学维度小于或等于$ 1 $（例如，$κ$是代数封闭的字段或有限字段）。令$ f $为$ k $以上的曲线的功能字段。让$ n $是一个不被char $（κ）$排除的无方体整数。然后，对于任何两个度$ n $ abelian的扩展，我们证明本地全球原则在离散估值方面适用于相关的多型圆环。令$ \ mathscr {x} $为$ f $的常规适当型号，使减少的特殊纤维$ x $是常规曲线的结合，具有正常的横梁。假设$κ$用$ char（κ）\ neq 2 $在代数上关闭。如果与$ \ Mathscr {x} $相关的图是一棵树（例如$ f = k（t）$），那么我们表明，相同的本地全球原理适用于与许多Abelian Extensions相关的MultiNorm torus，其中一种扩展是Quadratic的一个Quadratic，而其他延伸是不可分割的，而不是$ 4 $。
Let $K$ be a complete discretely valued field with the residue field $κ$. Assume that cohomological dimension of $κ$ is less than or equal to $1$ (for example, $κ$ is an algebraically closed field or a finite field). Let $F$ be the function field of a curve over $K$. Let $n$ be a squarefree positive integer not divisible by char$(κ)$. Then for any two degree $n$ abelian extensions, we prove that the local-global principle holds for the associated multinorm torus with respect to discrete valuations. Let $\mathscr{X}$ be a regular proper model of $F$ such that the reduced special fibre $X$ is a union of regular curves with normal crossings. Suppose that $κ$ is algebraically closed with $char(κ)\neq 2$. If the graph associated to $\mathscr{X}$ is a tree (e.g. $F = K(t)$) then we show that the same local-global principle holds for the multinorm torus associated to finitely many abelian extensions where one of the extensions is quadratic and others are of degree not divisible by $4$.
