Let $B$ be a quaternion algebra over $\mathbb{Q}$ and let $\mathcal{O} \subset B$ be an order.

A lattice in $B$ is (left) proper over $\mathcal{O}$ if its left order is equal to $\mathcal{O}$. We say $\mathcal{O}$ is good if every lattice in $B$ proper over $\mathcal{O}$ is invertible (equivalently, locally principal; equivalently, projective as a left $\mathcal{O}$-module).

How do you classify good quaternion orders $\mathcal{O}$?

If you ask only for containment--i.e., every lattice in $B$ with $\mathcal{O}$ contained in the left order is invertible--then you have the hereditary orders. If I haven't made a mistake, then there are Gorenstein orders that are not good.

I'm asking because every quadratic order has this property: every lattice in a quadratic field is invertible over its multiplicator ring--and this plays an important role in the theory of complex multiplication. I'm curious to see how far this holds for quaternions.

Any thoughts would be most appreciated!

Let $B$ be a quaternion algebra over $\mathbb{Q}$ and let $\mathcal{O} \subset B$ be an order.

A lattice in $B$ is (left) proper over $\mathcal{O}$ if its left order is equal to $\mathcal{O}$. We say $\mathcal{O}$ is good if every lattice in $B$ proper over $\mathcal{O}$ is invertible (equivalently, locally principal; equivalently, projective as a left $\mathcal{O}$-module).

How do you classify good quaternion orders $\mathcal{O}$?

If you ask only for containment--i.e., every lattice in $B$ with $\mathcal{O}$ contained in the left order is invertible--then you have the hereditary orders. If I haven't made a mistake, then there are Gorenstein orders that are not good.

I'm asking because every quadratic order has this property: every lattice in a quadratic field is invertible over its multiplicator ring--and this plays an important role in the theory of complex multiplication. I'm curious to see how far this holds for quaternions.

Any thoughts would be most appreciated!