Between the genus and the $\varGamma $-genus of an integral quadratic $\varGamma $-form
Volume 181 / 2017
Acta Arithmetica 181 (2017), 173-183
MSC: 11G20, 14H05, 11R29, 11R58, 11R65, 11F75.
DOI: 10.4064/aa8653-7-2017
Published online: 17 November 2017
Abstract
Let $\Gamma$ be a finite group and $(V,q)$ 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 to $(V,q)$ in the flat topology, and compare the genus $c(q)$ and the $\Gamma$-genus $c_\Gamma(q)$ of $q$. We show that $c_\Gamma(q)$ may not inject in $c(q)$. The obstruction comes from the failure of the Witt cancellation theorem for $\mathcal{O}_S$.
