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