1 results
Spinor groups with good reduction
- Part of
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- Journal:
- Compositio Mathematica / Volume 155 / Issue 3 / March 2019
- Published online by Cambridge University Press:
- 07 March 2019, pp. 484-527
- Print publication:
- March 2019
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- Article
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Let
$K$ be a two-dimensional global field of characteristic 
$\neq 2$ and let 
$V$ be a divisorial set of places of 
$K$. We show that for a given 
$n\geqslant 5$, the set of 
$K$-isomorphism classes of spinor groups 
$G=\operatorname{Spin}_{n}(q)$ of nondegenerate 
$n$-dimensional quadratic forms over 
$K$ that have good reduction at all 
$v\in V$ is finite. This result yields some other finiteness properties, such as the finiteness of the genus 
$\mathbf{gen}_{K}(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups 
$H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for 
$i\geqslant 1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type 
$\mathsf{G}_{2}$.
