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

Rosengarten, Zev

doi:10.2140/ant.2021.15.1865

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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

DOI: 10.2140/ant.2021.15.1865

Abstract

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.

Keywords

Tamagawa numbers, linear algebraic groups, pseudoreductive groups, Tate–Shafarevich sets

Mathematical Subject Classification 2010

Primary: 11R58

Secondary: 11R56, 11R34, 11E99

Milestones

Received: 27 June 2018

Revised: 16 January 2021

Accepted: 15 February 2021

Published: 10 November 2021

Authors

Zev Rosengarten
Einstein Institute of Mathematics Edmond J. Safra Campus The Hebrew University of Jerusalem Jerusalem Israel

Vol. 15, No. 8, 2021