Holomorphic map from the complex plane to a one-dimensional complex torus
= Holomorphic map from the complex plane to a one-dimensional complex torus
Every <holomorphic map> $f:\mathbb C\to\mathbb C/\Lambda$ is constant or surjective. It lifts to an entire function $F:\mathbb C\to\mathbb C$. If $F$ is nonconstant, the <Little Picard theorem> says it omits at most one point, while every fiber of $\mathbb C\to\mathbb C/\Lambda$ is an infinite lattice coset; hence every point of the torus has a preimage under $f$.