Failure of Weak Approximation in Adjoint Groups

TL;DR

This paper proves that adjoint groups do not satisfy weak approximation over arbitrary infinite fields, disproving a long-standing conjecture related to their rationality and approximation properties.

Contribution

It demonstrates the failure of weak approximation in adjoint groups, resolving a question that remained open after Merkurjev's disproof of their rationality.

Findings

Adjoint groups do not satisfy weak approximation over arbitrary infinite fields.

The conjecture by Platonov about weak approximation in adjoint groups is false.

The rationality of adjoint groups over infinite fields is not necessary for weak approximation.

Abstract

Platonov in 1991 conjectured that adjoint groups are rational as varieties over arbitrary infinite fields, and as a consequence have weak approximation. The rationality part of the conjecture was disproved by Merkurjev in 1996, but the question about weak approximation remained open. We settle this in the negative.