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

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 , $\mathcal {O}$...

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Bibliographic Details
Published in:Canadian journal of mathematics 2024-11, p.1-28
Main Authors: Goren, Eyal Z., Love, Jonathan R.
Format: Article
Language:English
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Summary: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.
ISSN:0008-414X
1496-4279
DOI:10.4153/S0008414X24000592