Brandt matrix - Knowlpedia

Let $\mathcal O$ be a maximal order in a definite quaternion algebra over $\mathbb Q$ , and choose representatives $I_1,\ldots,I_h$ of its right ideal classes. For $n\ge1$ , its Brandt matrix is

T(n)_{ij} =\#\{J\subseteq I_j:[I_j:J]=n^2,\ [J]=[I_i]\},

where $J$ ranges over locally principal right $\mathcal O$ -ideals. The index is that of additive lattices, and $[J]=[I_i]$ means $J=\alpha I_i$ for some invertible quaternion $\alpha$ . Column $j$ records the classes of the subideals of $I_j$ .

Prime neighbors

If $\ell$ is prime and does not divide the quaternion discriminant, every column of $T(\ell)$ sums to $\ell+1$ . It describes a directed multigraph on the ideal classes: the entry counts arrows from the column class to the row class, so multiple arrows are allowed. The matrix need not be symmetric in this basis.

Modular interpretation

These counts can also be expressed through representation numbers of positive definite quadratic lattices. Their theta series connect quaternion ideal arithmetic with weight-two modular forms. Transposing the convention exchanges row and column descriptions without changing the underlying correspondence.

References

John Voight, Quaternion Algebras, §41.1, equation (41.1.1), the following prime column-sum statement, and §§41.1.3–41.1.9.