  A fake quadric is a smooth projective surface that has the same rational
cohomology as a smooth quadric surface but is not biholomorphic to one. We
provide an explicit classification of all irreducible fake quadrics according
to the commensurability class of their fundamental group. To accomplish this
task, we develop a number of new techniques that explicitly bound the
arithmetic invariants of a fake quadric and more generally of an arithmetic
manifold of bounded volume arising from a form of SL_2 over a number field.
