Spinor Groups with Good Reduction

Let $K$ be a 2-dimensional global field of characteristic $\mathrm{\ne }2$, and let $V$ be a divisorial set of places of $K$. We show that for a given $n \geqslant 5$,thesetof$K$-isomorphismclassesofspinorgroups$G = $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}$_n(q)$ofnondegenerate$n$-dimensionalquadraticformsover$K$thathavegoodreductionatall$v \in V$, is finite. This result yields some other finiteness properties, such as the finiteness of the genus ${\mathbf{g}\mathbf{e}\mathbf{n}}_K(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups $H^i(K,{\mu }_2{)}_V$ for $i⩾1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type ${\mathsf{\text{G}}}_2$.