Rationally Isomorphic Hermitian Forms and Torsors of Some Non-Reductive Groups

TL;DR

This paper proves that under certain conditions, rational isomorphism of hermitian and quadratic forms over a semilocal Dedekind domain implies actual isomorphism, with connections to non-reductive group schemes and Bruhat--Tits theory.

Contribution

It establishes isomorphism criteria for hermitian and quadratic forms over non-reductive group schemes, extending classical results and relating to the Grothendieck--Serre conjecture.

Findings

Rationally isomorphic hermitian forms are isomorphic under certain conditions.

Quadratic forms with finite group actions exhibit similar isomorphism properties.

Connections to Bruhat--Tits theory for non-reductive group schemes.

Abstract

Let $R$ be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an $R$-algebra with involution, which are rationally ismorphic and have isomorphic semisimple coradicals, are in fact isomorphic. The same result is also obtained for quadratic forms equipped with an action of a finite group. The results have cohomological restatements that resemble the Grothendieck--Serre conjecture, except the group schemes involved are not reductive. We show that these group schemes are closely related to group schemes arising in Bruhat--Tits theory.