  It is a longstanding problem to determine the precise relationship between
the geodesic length spectrum of a hyperbolic manifold and its commensurability
class. A well known result of Reid, for instance, shows that the geodesic
length spectrum of an arithmetic hyperbolic surface determines the surface's
commensurability class. It is known, however, that non-commensurable arithmetic
hyperbolic surfaces may share arbitrarily large portions of their length
spectra. In this paper we investigate this phenomenon and prove a number of
quantitative results about the maximum cardinality of a family of pairwise
non-commensurable arithmetic hyperbolic surfaces whose length spectra all
contain a fixed (finite) set of nonnegative real numbers.
