Computing ideals of a given norm in Quaternion algebra

asked 2022-08-19 04:10:16 +0100

DrewC gravatar image

updated 2022-08-19 09:00:52 +0100

slelievre gravatar image

Let $B$ be the quaternion algebra over $\mathbb{Q}$ ramified at prime $p$ and $\infty$ and $\mathcal{O}\subset B$ be a maximal order. How do you compute all integral right ideals $ I \subseteq \mathcal{O}$ of a given norm?

Say $l$ is prime. You know BrandtModule(p).hecke_matrix(l) returns $l$-Brandt matrix of B and we can draw the $l$-Brandt graph from it, then $\mathcal{O}$ is a vertex and there are $l+1$ integral right $\mathcal{O}$-ideals of norm $l$ representing each of edge from $\mathcal{O}$ in general. I'd like to compute these $l$-neighbors of $\mathcal{O}$.

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