# Pathological Behavior of Arithmetic Invariants of Unipotent Groups

## TL;DR

This paper demonstrates that many favorable properties of arithmetic invariants like Tamagawa numbers for pseudo-reductive groups do not extend to non-commutative unipotent groups, highlighting their pathological behavior.

## Contribution

It reveals the failure of expected arithmetic properties in non-commutative unipotent groups and provides some positive results for connected linear algebraic groups.

## Key findings

- Arithmetic invariants behave pathologically in non-commutative unipotent groups.
- Tamagawa numbers show some reasonable behavior in broader classes of algebraic groups.
- Failure of nice properties previously established for pseudo-reductive groups.

## Abstract

We show that all of the nice behavior for Tamagawa numbers, Tate-Shafarevich sets, and other arithmetic invariants of pseudo-reductive groups over global function fields proved in \cite{rospred} fails in general for non-commutative unipotent groups. We also give some positive results which show that Tamagawa numbers do exhibit some reasonable behavior for arbitrary connected linear algebraic groups over global function fields.