AI 中文总结
研究函数域上乘法型群的泰特 - 沙法列维奇群的有限性，通过对\(k\)的条件设定（有限生成且\(X(k)\neq\emptyset\)或\(k\)是数域），证明相应泰特 - 沙法列维奇群有限，补充了\(X\)是曲线情形的相关工作。
AI 中文摘要
设\(K = k(X)\)是特征为\(0\)的域\(k\)上维度\(\geq 2\)的光滑几何整簇\(X\)的函数域，\(V\)是\(K\)中与\(X\)上素除子相关的离散赋值集。我们证明，若\(D\)是\(k\)定义的乘法型群，那么在以下情形下，相应的泰特 - 沙法列维奇群\(Sha(D,V)=\ker(H^1(K,D)\to\prod_{v\in V}H^1(K_v,D))\)是有限的：(1)\(k\)是有限生成且\(X(k)\neq\emptyset\)；(2)\(k\)是数域。这补充了哈拉里和萨穆埃利之前关于\(X\)是曲线情形的工作。
英文摘要
Let $K = k(X)$ be the function field of a smooth geometrically integral variety $X$ of dimension $\geq 2$ over a field $k$ of characteristic 0 and $V$ be the set of discrete valuations of $K$ associated with the prime divisors on $X$. We show that if $D$ is a $k$-defined group of multiplicative type, then the corresponding Tate-Shafarevich group $Sha(D,V) = \ker \left(H^1(K,D) \to \prod_{v \in V} H^1(K_v, D) \right)$ is finite in the following situations: (1) $k$ is finitely generated and $X(k) \neq \emptyset$; (2) $k$ is a number field. This complements previous work of Harari and Szamuely, which considered the case where $X$ is a curve.
