Adjoint groups over ${\mathbb Q}_p (X)$ and R-equivalence - revisited

TL;DR

This paper constructs examples of non-rational adjoint classical groups over function fields of curves over p-adic fields and proves triviality of R-equivalence classes for groups of type C_n over these fields.

Contribution

It provides new examples of non-rational adjoint groups of specific types over function fields and establishes R-triviality for groups of type C_n in this context.

Findings

Constructed non-rational adjoint groups of types ^2A_n and ^2D_3 over function fields.

Proved R-equivalence classes are trivial for groups of type C_n over these fields.

Enhanced understanding of rationality and R-equivalence over p-adic function fields.

Abstract

We obtain a class of examples of non-rational adjoint classical groups of type $^2A_n$ and a group of type $^2D_3$ over the function field $F$ of a smooth geometrically integral curve over a $p$-adic field with $p \neq 2$. We also show that for any group of type $C_n$ over $F$, the group of rational equivalence classes of $G$ over $F$ is trivial, i.e., $G(F)/R=(1)$.