On elements of prescribed norm in maximal orders of a quaternion algebra

Abstract Let $\mathcal {O}$ be a maximal order in the quaternion algebra over $\mathbb Q$ ramified at p and $\infty $ . We prove two theorems that allow us to recover the structure of $\mathcal {O}$ from limited information. The first says that for any infinite set S of integers coprime to p , $\mathcal {O}$ is spanned as a ${\mathbb {Z}}$ -module by elements with norm in S . The second says that $\mathcal {O}$ is determined up to isomorphism by its theta function.