Properness of the global-to-local map for algebraic groups with toric connected component and other finiteness properties

IF 0.6 3区 数学 Q3 MATHEMATICS Mathematical Research Letters Pub Date : 2023-12-15 DOI:10.4310/mrl.2023.v30.n3.a11
Andrei S. Rapinchuk, Igor A. Rapinchuk
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引用次数: 0

Abstract

This is a companion paper to $\href{ https://doi.org/10.1016/j.jnt.2021.07.001}{[29]}$, where we proved the finiteness of the Tate–Shafarevich group for an arbitrary torus $T$ over a finitely generated field $K$ with respect to any divisorial set $V$ of places of $K$. Here, we extend this result to any $K$-group $D$ whose connected component is a torus (for the same $V$), and as a consequence obtain a finiteness result for the local-to-global conjugacy of maximal tori in reductive groups over finitely generated fields. Moreover, we prove the finiteness of the Tate–Shafarevich group for tori over function fields $K$ of normal varieties defined over base fields of characteristic zero and satisfying Serre’s condition (F), in which case $V$ consists of the discrete valuations associated with the prime divisors on the variety (geometric places). In this situation, we also establish the finiteness of the number of $K$-isomorphism classes of algebraic $K$-tori of a given dimension having good reduction at all $v \in V$ , and then discuss ways of extending this result to positive characteristic. Finally, we prove the finiteness of the number of isomorphism classes of forms of an absolutely almost simple group defined over the function field of a complex surface that have good reduction at all geometric places.
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具有环状连通分量的代数群的全局到局部映射的适当性及其他有限性性质
本文是$\href{ https://doi.org/10.1016/j.jnt.2021.07.001}{[29]}$ 的姊妹篇,我们在其中证明了有限生成域$K$上任意环$T$的塔特-沙法列维奇群的有限性,它与$K$的任意分位集$V$有关。在这里,我们将这一结果扩展到任何其连接部分是环的(对于相同的 $V$)$K$-群 $D$,并由此获得了有限生成域上还原群中最大环的局部到全局共轭的有限性结果。此外,我们还证明了定义在特征为零的基域上并满足塞雷条件(F)的正态变体的函数域 $K$ 上的转矩的塔特-沙法列维奇群的有限性,在这种情况下,$V$ 包含与变体上的素除数相关的离散值(几何位置)。在这种情况下,我们还建立了在 V$ 中所有 $v \ 处都有良好还原的给定维数代数 $K$ 托里的 $K$ 同构类的有限性,然后讨论了将这一结果扩展到正特征的方法。最后,我们证明了在复曲面的函数域上定义的绝对近简群形的同构类的有限性,这些同构类在所有几何位置上都有良好的还原。
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来源期刊
CiteScore
1.40
自引率
0.00%
发文量
9
审稿时长
6.0 months
期刊介绍: Dedicated to publication of complete and important papers of original research in all areas of mathematics. Expository papers and research announcements of exceptional interest are also occasionally published. High standards are applied in evaluating submissions; the entire editorial board must approve the acceptance of any paper.
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