We study Tamagawa numbers and other invariants (especially Tate–Shafarevich sets) attached to commutative and pseudoreductive groups over global function fields. In particular, we prove a simple formula for Tamagawa numbers of commutative groups and pseudoreductive groups. We also show that the Tamagawa numbers and Tate–Shafarevich sets of such groups are invariant under inner twist, as well as proving a result on the cohomology of such groups which extends part of classical Tate duality from commutative groups to all pseudoreductive groups. Finally, we apply this last result to show that for suitable quotient spaces by commutative or pseudoreductive groups, the Brauer–Manin obstruction is the only obstruction to strong (and weak) approximation.

我们研究 Tamagawa 数和其他不变量（尤其是 Tate-Shafarevich 集）附加到全局函数域上的交换和伪还原群。特别是，我们证明了交换群和伪还原群的玉川数的简单公式。我们还表明，此类群的 Tamagawa 数和 Tate-Shafarevich 集在内扭曲下是不变的，并证明了此类群的上同调的结果，该结果将部分经典 Tate 对偶性从交换群扩展到所有伪还原群。最后，我们应用最后一个结果来表明对于交换或伪还原群的合适商空间，Brauer-Manin 障碍是强（和弱）逼近的唯一障碍。