On elliptic curves in {${\rm SL}\sb 2(\Bbb C)/\Gamma$}, Schanuel's conjecture and geodesic lengths

Abstract

Let $\Gamma$ be a discrete cocompact subgroup of $\SL_{2}(\C)$. We conjecture that the quotient manifold $X = \SL_{2}(\C)/\Gamma$ 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 $\Gamma \cap \SL_{2}(\R)$ is cocompact in $\SL_{2}(\R)$.

Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic $2$- and $3$-folds.