# Between the genus and the $\Gamma$-genus of an integral quadratic $\Gamma$-form

## TL;DR

This paper investigates the classification of quadratic forms with group actions over global function fields, revealing differences between the genus and the mma-genus due to the failure of Witt cancellation.

## Contribution

It introduces a cohomological approach to classify mma-forms and compares their genus and mma-genus, highlighting the non-injectivity caused by Witt cancellation failure.

## Key findings

- mma-genus does not inject into the genus.
- Witt cancellation failure affects classification.
- Cohomological methods classify forms over mbda-forms.

## Abstract

Let \Gamma be a finite group and (V,q) be a regular quadratic \Gamma-form defined over an integral domain $\mathcal{O}_S$ of a global function field (of odd characteristic). We use flat cohomology to classify the quadratic \Gamma-forms defined over $\mathcal{O}_S$ that are locally \Gamma-isomorphic for the flat topology to (V,q) and compare between the genus c(q) and the \Gamma-genus c_{\Gamma}(q) of q. We show that c_{\Gamma}(q) should not inject in c(q). The suggested obstruction arises from the failure of the Witt cancellation theorem for $\mathcal{O}_S$.