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

TL;DR

This paper investigates Tamagawa numbers and related invariants of pseudo-reductive groups over global function fields, providing formulas, invariance results, and applications to rational points and obstructions.

Contribution

It establishes a simple formula for Tamagawa numbers, proves their invariance under inner twist, and extends Tate duality to pseudo-reductive groups.

Findings

Derived a formula for Tamagawa numbers of pseudo-reductive groups.

Proved invariance of Tamagawa numbers and Tate-Shafarevich sets under inner twist.

Showed Brauer-Manin obstruction is the only obstruction for certain quotient spaces.

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.