Abstract
In a recent article with Preeti R., Archita Mondal established rational equivalence for semi-simple adjoint classical groups over specific fields, building on Manin's foundational work from 1974. They explored the relationship between the rational equivalence of G(E), the group of E-rational points of G, and the rationality of the underlying group variety G over a field E.
This project aims to construct examples of non-rational adjoint groups that remain underexplored. A key unresolved question concerns the finiteness of the group of rational equivalence classes, G(E)/R. While P. Gille demonstrated its finiteness over the function field of complex surfaces, we seek to identify a counterexample where G(E)/R is infinite for a specific group and field. We will leverage Merkurjev's characterization of G(E)/R in terms of multipliers of similitudes in central simple algebras with involution as our primary tool.
Additionally, we aim to address the norm principle, introduced by Serre, which has been studied extensively by Gille and Merkurjev. While the principle generally holds for reductive classical groups, exceptions arise in Dynkin type D. In 2016, N. Bhaskar established the norm principle for type D using a 'special' element, and by 2019, it was extended to split central simple algebras by Bhaskar, Merkurjev, and Chernousov. Our goal is to further extend these results to the non-split case, ultimately resolving the norm principle for reductive classical groups of type D.
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