  Let H be a discrete cocompact subgroup of SL_2(C). We conjecture that the
quotient manifold X=SL_2(C)/H contains infinitely many non-isogeneous elliptic
curves and prove that this is indeed the case if Schanuel's conjecture holds.
We also prove it in the special case where the intersection of H and SL_2(R) is
cocompact in SL_2(R).
  Furthermore, we deduce some consequences for the geodesic length spectra of
real hyperbolic 2- and 3-folds.
