  We apply G. Prasad's volume formula for the arithmetic quotients of
semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic
subgroups of SO(1,n). As a result we prove that for any even dimension n there
exists a unique compact arithmetic hyperbolic n-orbifold of the smallest
volume. We give a formula for the Euler-Poincare characteristic of the
orbifolds and present an explicit description of their fundamental groups as
the stabilizers of certain lattices in quadratic spaces. We also study
hyperbolic 4-manifolds defined arithmetically and obtain a number theoretical
characterization of the smallest compact arithmetic 4-manifold.
