= Affine lift of a holomorphic map between one-dimensional complex tori
= Holomorphic lift between one-dimensional complex tori
{synonym}
Let $\pi_\Lambda:\mathbb C\to\mathbb C/\Lambda$ and $\pi_{\Lambda'}:\mathbb C\to\mathbb C/\Lambda'$ be the <universal covering maps> of two one-dimensional <complex tori>. Every <holomorphic map> $f:\mathbb C/\Lambda\to\mathbb C/\Lambda'$ has a <holomorphic lift between one-dimensional complex tori> $F:\mathbb C\to\mathbb C$ satisfying $\pi_{\Lambda'}\circ F=f\circ\pi_\Lambda$, and every such lift is an <affine function>
$$
F(z)=az+b
$$
with $a\Lambda\subseteq\Lambda'$. Indeed, for $\lambda\in\Lambda$, the continuous function $F(z+\lambda)-F(z)$ takes values in the discrete set $\Lambda'$ and is therefore constant. Hence $F'$ is $\Lambda$-periodic. It is bounded on a compact <fundamental parallelogram> and therefore on the <complex plane>, so the <Liouville theorem> makes $F'$ constant. If $f$ preserves the identity and the lift is chosen with $F(0)=0$, then $b=0$ and the lift is a <linear map>.
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