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

Rony A. Bitan¹

¹ Department of Mathematics Bar-Ilan University Ramat-Gan 5290002, Israel and Afeka, Tel-Aviv Academic College of Engineering Tel-Aviv 6910717, Israel

Acta Arithmetica, Tome 181 (2017) no. 2, pp. 173-183

Résumé

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 $\mathcal{O}_S$.

Détail

DOI : 10.4064/aa8653-7-2017

Keywords: gamma finite group regular quadratic gamma form defined integral domain mathcal global function field odd characteristic flat cohomology classify quadratic gamma forms defined mathcal locally gamma isomorphic flat topology compare genus gamma genus gamma gamma may inject obstruction comes failure witt cancellation theorem nbsp mathcal