Conjugacy classes of $SU(h,\mathcal O_S)$ in $SL(2,\mathcal O_S)$

Abstract:

Let $K$ be a quadratic extension of a global field $F$, of characteristic not two, and $\mathcal {O}_S$ the integral closure in $K$ of a Dedekind ring of $S$-integers $\mathfrak {O}_S$ in $F$. Then $PSL(2, \mathcal {O}_S)$ is isomorphic to the spinorial kernel $O’(L)$ for an indefinite quadratic $\mathfrak {O}_S$-lattice $L$ of rank 4. The isomorphism is used to study the conjugacy classes of unitary groups $PSU(h,\mathcal {O}_S)$ of primitive odd binary hermitian matrices $h$ under the action of $PSL(2, \mathcal {O}_S)$.