Holomorphic flexibility properties of spaces of elliptic functions

Let $X$ be an elliptic curve and $mathbb{P}$ the Riemann sphere. Since $X$ is compact, it is a deep theorem of Douady that the set $mathcal{O}(X,mathbb{P})$ consisting of holomorphic maps $Xto mathbb{P}$ admits a complex structure. If $R_n$ denotes the set of maps of degree $n$, then Namba has shown for $ngeq2$ that $R_n$ is a $2n$-dimensional complex manifold. We study holomorphic flexibility properties of the spaces $R_2$ and $R_3$. Firstly, we show that $R_2$ is homogeneous and hence an Oka manifold. Secondly, we present our main theorem, that there is a $6$-sheeted branched covering space of $R_3$ that is an Oka manifold. It follows that $R_3$ is $mathbb{C}$-connected and dominable. We show that $R_3$ is Oka if and only if $mathbb{P}_2backslash C$ is Oka, where $C$ is a cubic curve that is the image of a certain embedding of $X$ into $mathbb{P}_2$. We investigate the strong dominability of $R_3$ and show that if $X$ is not biholomorphic to $mathbb{C}/Gamma_0$, where $Gamma_0$ is the hexagonal lattice, then $R_3$ is strongly dominable. As a Lie group, $X$ acts freely on $R_3$ by precomposition by translations. We show that $R_3$ is holomorphically convex and that the quotient space $R_3/X$ is a Stein manifold. We construct an alternative $6$-sheeted Oka branched covering space of $R_3$ and prove that it is isomorphic to our first construction in a natural way. This alternative construction gives us an easier way of interpreting the fibres of the branched covering map.