On Elliptic Curves in SL_2(C)/\Gamma, Schanuel's conjecture and geodesic lengths

Let H be a discrete cocompact subgroup of SL_2©. We conjecture that the quotient manifold X=SL_2©/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® is cocompact in SL_2®. Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic 2- and 3-folds.