  We apply a study of orders in quaternion algebras, to the differential
geometry of Riemann surfaces. The least length of a closed geodesic on a
hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P.
Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in
the genus g(X). The Fuchsian groups in their examples are principal congruence
subgroups of a fixed arithmetic group with rational trace field. We generalize
their construction to principal congruence subgroups of arbitrary arithmetic
surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in
a quaternion algebra. We obtain a particularly sharp bound for a principal
congruence tower of Hurwitz surfaces (PCH), namely the 4/3-bound
syspi_1(X_{\PCH}) > 4/3 \log(g(X_{\PCH})). Similar results are obtained for the
systole of hyperbolic 3-manifolds, relative to their simplicial volume.
