Rotations and units in quaternion algebras (Q412108)

Let \(K\) be a number field with ring of integers \(R\), and let \(A=\left( \frac{a,b}{K}\right)\) be a quaternion algebra over \(K\). To calculate the unit group of an order \(\mathcal O\) in \(A\), the group \(\mathcal O^1\) of elements of reduced norm 1 plays a decisive rôle. This leads to the Diophantine equation NEWLINE\[NEWLINEx^2-ay^2-bz^2+abt^2=1,NEWLINE\]NEWLINE a quaternionic analogue of the Pell equation. The group \(\mathcal O^1\) can be regarded as an arithmetic group of \(\mathrm{SL}_2(\mathbb C)\). This leads to an action of \(\mathcal O^1\) on the hyperbolic space \(H^3\). There are six cases where this action is discontinuous, one of them with \(K\) totally real and \(A\) ramifying at all but one real embeddings of \(K\), for example, if \(a=b=-1\) and \(Z(\mathcal O)=R\) for an imaginary quadratic field \(K\). Such an algebra \(A\) is a skew-field if and only if \(K=\mathbb Q(\sqrt{d})\) with \(d\equiv 1 \pmod 8\). Using the short exact sequence NEWLINE\[NEWLINE1\to K^\times\to A^\times\to \mathrm{SO}_3(K)\to 1,NEWLINE\]NEWLINE the unit group of \(\mathcal O\) is studied in this particular case.