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

TL;DR

This paper investigates how the structure of a maximal order in a quaternion algebra over rationals can be reconstructed from limited norm information and its theta function, advancing understanding of quaternion algebra invariants.

Contribution

It proves that the maximal order can be recovered from elements with norms in any infinite coprime set and is uniquely determined by its theta function.

Findings

Maximal order spanned by elements with norms in any infinite coprime set.

Maximal order uniquely determined by its theta function.

Provides methods to recover algebraic structure from limited norm data.

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.