Orthogonal representations of affine group schemes and twists of symmetric bundles

TL;DR

This paper extends the theory of twisting quadratic forms by group scheme torsors to affine schemes with non-constant group schemes, utilizing Hopf-algebra quadratic theory to handle ramification.

Contribution

It introduces a framework for understanding twists of quadratic forms by non-constant affine group schemes, including ramified torsors, via Hopf-algebra methods.

Findings

Develops a new approach for non-constant group schemes over affine bases.

Incorporates ramification effects into the twist theory.

Provides formulas for Hasse-Witt invariants in this generalized setting.

Abstract

Following Serre's initial work, a number of authors have considered twists of quadratic forms on a scheme Y by torsors of a finite group G, together with formulas for the Hasse-Witt invariants of the twisted form. In this paper we take the base scheme Y to be affine and consider non-constant groups schemes G. There is a fundamental new feature in this case - in that the torsor may now be ramified over Y. The natural framework for handling the case of a non-constant group scheme over the affine base is provided by the quadratic theory of Hopf-algebras.