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Let $p \neq 2$ and let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$ and maximal ideal $\mathfrak{p}$. By a quadratic space $V_{\mathcal{O}}$ of dimension $d$ over $\mathcal{O}$, I mean ...

Let $n \geq 3$. Let $A_0$ denote the $n \times n$ symmetric matrix with $1$'s on the antidiagonal and $0$'s everywhere else. We can define the associated special orthogonal group $$ \mathrm{SO}(A_0, \...

Let $O(2,\mathbb{Z}_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for ...