Let $H$ be a quaternionic algebra over ${\bf Q}$, and let $R$ denote a maximal ${\bf Z}$-order in $H$. Is there a formula for the rank of the (abelianization of the) units in $R$ analogous to the Dirichlet unit theorem? Is there an analogous formula for the $S$-units?

Let $H$ be a quaternionic algebra over ${\bf Q}$, and let $R$ denote a maximal ${\bf Z}$-order in $H$. Is there a theorem on the structure of the units in $R$ analogous to the Dirichlet unit theorem? Is there an analogous theorem for the $S$-units?

Let $H$ be a quaternionic algebra over ${\bf Q}$, and let $R$ denote a maximal ${\bf Z}$-order in $H$. Is there a formula for the rank of the units in $R$ analogous to the Dirichlet unit theorem? Is there an analogous formula for the $S$-units?

Let $H$ be a quaternionic algebra over ${\bf Q}$, and let $R$ denote a maximal ${\bf Z}$-order in $H$. Is there a formula for the rank of the (abelianization of the) units in $R$ analogous to the Dirichlet unit theorem? Is there an analogous formula for the $S$-units?