I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the (non-degenerate) quadratic forms on the $3$-dimensional $p$-adic vector space ${\mathbb Q}\_p^3$, one compact and one non-compact. The reference provided is Serre's "a course in arithmetic", Chapter IV, in which I was not able to find a proof. As far as I know, up to equivalence and scaling, there are three anisotropic and three isotropic such quadratic forms. Is it obvious that the corresponding orthogonal groups in each class are isomorphic?



$p$-adic analogues of $\mathrm{SO}(3)$
If $q$ is a quadratic form over a field $k$, the orthogonal group $\mathrm{O}(q)$ doesn't change if you replace $q$ by $\lambda q$ for $\lambda\in k^\*$. In rank 3, since $\lambda ^3\equiv \lambda $ (mod. $k^{\*2}$), this implies that you can consider only forms of discriminant 1. Over $\mathbb{Q}\_p$ $(p>2)$ this leaves only the invariant $\varepsilon $ (defined in terms of the Hilbert symbol), equal to $\pm 1$.


