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arXiv · 2307.16828

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

Eyal Z. Goren·Jonathan R. Love
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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.

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Eyal Z. Goren, Jonathan R. Love. 2023-07-31. On elements of prescribed norm in maximal orders of a quaternion algebra. https://doi.org/10.4153/s0008414x24000592

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