Reduced discriminant of a quaternion Z-order

Let $\mathcal O$ be a $\mathbb Z$ -order in a quaternion algebra $B/\mathbb Q$ , and let $e_1,\ldots,e_4$ be a $\mathbb Z$ -basis. Its reduced discriminant is the positive integer

\operatorname{discrd}(\mathcal O)= \sqrt{\left|\det\bigl(\operatorname{trd}(e_i e_j)\bigr)_{i,j=1}^4\right|},

where $\operatorname{trd}$ is reduced trace. For a quaternion order the determinant has square absolute value, and the result is independent of the integral basis.

Basis changes and larger orders

A unimodular basis change multiplies the determinant by the square of $\pm1$ , hence leaves it unchanged. If $\mathcal O\subseteq\mathcal O'$ , then

\operatorname{discrd}(\mathcal O) =[\mathcal O':\mathcal O]\operatorname{discrd}(\mathcal O').

The positive integer is therefore distinct from the determinant itself, whose absolute value is its square.

Lipschitz and Hurwitz orders

For the Lipschitz order, the basis $1,i,j,k$ has trace-pairing matrix $\operatorname{diag}(2,-2,-2,-2)$ , so the reduced discriminant is $4$ .

The Hurwitz order contains it with index two, so its reduced discriminant is $2$ . This equals the discriminant of the rational Hamilton algebra. In general, equality with the algebra discriminant characterizes maximal quaternion $\mathbb Z$ -orders.

References

John Voight, Quaternion Algebras, author's text, §15.1, Definition 15.4.4, Lemma 15.4.7, and Theorem 15.5.5.