阿代尔点与非分歧布饶尔逼近在分类栈中的应用
Adelic points and unmramified Brauer approximation for classifying stacks
AI总结：
本文针对分类栈BG的强逼近问题，对比三种方法并证明非分歧布饶尔群相关同构，给出非分歧布饶尔逼近的局部准则，揭示其失效情形。
AI中文摘要：
设k为数域，G为k上的连通线性代数群。我们针对分类栈BG关于其全布饶尔群的强逼近，对比三种方法：Dhillon的分类栈齐性空间方法、Kottwitz的局部-整体序列，以及约化群情形下Borovoi的局部化定理。在自然同构Br(BG)/Br(k)≅Pic(G)下，我们将Kottwitz与Borovoi的障碍映射识别为布饶尔赋值，因此三种方法对局部化像给出了与投影全布饶尔-马宁集一致的精确描述。随后我们研究关于普通非分歧布饶尔群的强逼近，证明Br^{un}(BG)/Br(k)≅Sha^1_{cyc}(k,Ĝ)（其中Ĝ=X^*(G_{\bar{k}})），并给出有限个位之外非分歧布饶尔逼近的精确局部准则。实例表明，即使省略一个有限位，该逼近也可能失效，且存在一个环面，其非分歧布饶尔群将整体像截为阿代尔空间的真子集。
英文摘要：
Let $k$ be a number field and let $G$ be a connected linear algebraic group over $k$. We compare three approaches to strong approximation for the classifying stack $BG$ with respect to its full Brauer group: the homogeneous-space method of \cite{DhillonClassifying}, Kottwitz's local--global sequence, and, for reductive groups, Borovoi's localization theorem. Under the natural identification \[ Br(BG)/Br(k)\simeq Pic(G), \] we identify the Kottwitz and Borovoi obstruction maps with Brauer evaluation. The three approaches therefore give the same exact description of the localization image as the projected full Brauer--Manin set. We then study strong approximation with respect to the ordinary unramified Brauer group. We prove \[ Br^{un}(BG)/ Br(k)\simeq Sha^1_{\mathrm{cyc}}(k,\widehat G), \qquad \widehat G=X^*(G_{\bar{k}}), \] and give an exact local criterion for unramified Brauer approximation off a finite set of places. Examples show that this approximation can fail even when a finite place is omitted, and exhibit a torus for which the unramified Brauer group cuts out the global image as a proper subset of the adelic space.
