Holomorphic flexibility properties of spaces of elliptic functions

Abstract

Let X be an elliptic curve and P the Riemann sphere. Since X is compact, it is a deep theorem of Douady that the set O(X, P) consisting of holomorphic maps X → P admits a complex structure. If R𝑛 denotes the set of maps of degree n, then Namba has shown for n ≥ 2 that R𝑛 is a 2n-dimensional complex manifold. We study holomorphic flexibility properties of the spaces R₂ and R₃. Firstly, we show that R₂ is homogeneous and hence an Oka manifold. Secondly, we present our main theorem, that there is a 6-sheeted branched covering space of R₃ that is an Oka manifold. It follows that R₃ is C-connected and dominable. We show that R₃ is Oka if and only if P₂\C is Oka, where C is a cubic curve that is the image of a certain embedding of X into P₂. We investigate the strong dominability of R₃ and show that if X is not biholomorphic to C/Γ₀, where Γ₀ is the hexagonal lattice, then R₃ is strongly dominable. As a Lie group, X acts freely on R₃ by precomposition by translations. We show that R₃ is holomorphically convex and that the quotient space R₃/X is a Stein manifold. We construct an alternative 6-sheeted Oka branched covering space of R₃ 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.