Rational equivalence on adjoint groups of type $^{1}D_n$ over field $\mathbb{Q}_P(X)$

TL;DR

This paper proves that for certain algebraic groups over the function field of a curve over a p-adic field, the rational equivalence classes are trivial, advancing understanding of algebraic group rationality over such fields.

Contribution

It establishes the triviality of rational equivalence classes for adjoint groups of type $^{1}D_n$ over specific function fields, a novel result in algebraic group theory.

Findings

$G(F)/R$ is trivial for the specified groups and fields

Advances understanding of rational equivalence in algebraic groups over function fields

Provides new insights into the structure of adjoint groups over $p$-adic function fields

Abstract

Let $F$ be the function field of a smooth, geometrically integral curve over a $p$-adic field with $p\neq 2.$ Let $G$ be a classical adjoint group of type $^1D_n$ defined over $F$. We show that $G(F) / R$ is trivial, where $R$ denotes {\it rational equivalence} on $G(F)$.