More Examples of Non-Rational Adjoint Groups

TL;DR

This paper introduces a recursive method to construct quadratic forms with non-stably rational adjoint groups, extending previous work by analyzing R-equivalence classes to demonstrate non-triviality.

Contribution

It provides a new recursive construction for quadratic forms in the Witt ring's fundamental ideal with non-stably rational adjoint groups, expanding known examples.

Findings

Constructed quadratic forms in the n-th power of the Witt ring's fundamental ideal.

Proved that the associated adjoint groups are not stably rational.

Demonstrated non-R-triviality of these groups using R-equivalence class computations.

Abstract

We give a recursive construction to produce examples of quadratic forms q_n in the n-th power of the fundamental ideal in the Witt ring whose corresponding adjoint groups PSO(q_n) are not stably rational. Computations of the R-equivalence classes of adjoint classical groups by A.S. Merkurjev are used to show that these groups are not R-trivial. This extends earlier results of A.S. Merkurjev and P.Gille where the forms considered have non-trivial and trivial discriminants respectively.