AI 中文总结
本文研究代数闭域上代数群的有限阿贝尔子群，深化其结构结果并应用于解决Totaro问题、证明Tits假设变体及挠子分裂相关结论。
AI 中文摘要
设k为代数闭域，G为代数k-群，本文研究阶数不被k的特征整除的有限阿贝尔k-子群A⊂G，这是可追溯至20世纪60年代初Borel工作的代数群理论经典课题。本文深化了关于A结构的已有结果，特别证明存在G的极大环面T，使得指数[A:(A∩T)整除Grothendieck挠指数t(G)；还证明存在极大环面T，使得商群A/(A∩T)在合适意义下“很小”。作为这些结果的应用，本文：(i) 对迭代Laurent级数域k_r = k((t₁))((t₂))…((t_r))上的G-挠子，正面回答了Totaro提出的问题；(ii) 证明了关于E₈-挠子分裂域的Tits“乐观假设”的一个变体；(iii) 证明了k_r上的某些挠子无法被亏格1曲线的函数域分裂。
英文摘要
Let $k$ be an algebraically closed field, and let $G$ be an algebraic $k$-group. We study finite abelian $k$-subgroups $A \subset G$ whose order is not divisible by the characteristic of $k$. This is a classical topic in the theory of algebraic groups going back to the work of Borel in the early 1960s. We sharpen previously known results on the structure of $A$. In particular, we show that there exists a maximal torus $T$ of $G$ such that the index $[A: (A \cap T)]$ divides the Grothendieck torsion index $t(G)$. We also show that there exists a maximal torus $T$ such that the quotient group $A/(A \cap T)$ is ``small'' in a suitable sense. As applications of these results, we (i) give a positive answer to a question of Totaro for $G$-torsors over fields $k_r = k((t_1))((t_2)) \ldots ((t_r))$ of iterated Laurent series, (ii) prove a variant of the ``hypothèse optimiste'' of Tits about splitting fields of $E_8$-torsors, and (iii) show that certain torsors over $k_r$ cannot be split by the function field of a genus $1$ curve.
Comments29 pages
