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arXiv · 1806.10723

Tamagawa Numbers and Other Invariants of Pseudo-reductive Groups Over Global Function Fields

Zev Rosengarten
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Abstract

We study Tamagawa numbers and other invariants (especially Tate-Shafarevich sets) attached to commutative and pseudo-reductive groups over global function fields. In particular, we prove a simple formula for Tamagawa numbers of commutative groups and pseudo-reductive 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 pseudo-reductive groups. Finally, we apply this last result to show that for suitable quotient spaces by commutative or pseudo-reductive groups, the Brauer--Manin obstruction is the only obstruction to strong (and weak) approximation.

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BibTeXRIS

Zev Rosengarten. 2020-01-31. Tamagawa Numbers and Other Invariants of Pseudo-reductive Groups Over Global Function Fields. https://doi.org/10.2140/ant.2021.15.1865

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