Generating sets for maximal orders in rational quaternion algebras

Abstract

Given a maximal order $\mathfrak{O}$ in a rational definite quaternion algebra, and a prime $\ell$ coprime to the discriminant of $\mathfrak{O}$, this paper considers subsets of $\mathfrak{O}$ consisting of elements with $\ell$-power norms that together generate $\mathfrak{O}$ as a $\mathbb{Z}$-algebra. We prove two theorems about the existence of such sets: the first states that $\mathfrak{O}$ is generated by elements of norm $\ell^k$ for any $k$ larger than an explicit bound, and the second states that there is a generating set for $\mathfrak{O}$ consisting of at most three elements, each with norm a power of $\ell$. We discuss implications for the study of supersingular isogeny graphs. As steps towards these theorems, for quaternion orders $\mathcal{O}$ that are not necessarily maximal, we also prove structural results about the algebra of Brandt matrices for $\mathcal{O}$ and explicit bounds on the coefficients of the theta function of $\mathcal{O}$. Computational experiments are also discussed.