Vol. 15, No. 8, 2021

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Tamagawa numbers and other invariants of pseudoreductive groups over global function fields

Zev Rosengarten

Vol. 15 (2021), No. 8, 1865–1920

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.

Tamagawa numbers, linear algebraic groups, pseudoreductive groups, Tate–Shafarevich sets
Mathematical Subject Classification 2010
Primary: 11R58
Secondary: 11R56, 11R34, 11E99
Received: 27 June 2018
Revised: 16 January 2021
Accepted: 15 February 2021
Published: 10 November 2021
Zev Rosengarten
Einstein Institute of Mathematics
Edmond J. Safra Campus
The Hebrew University of Jerusalem