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Given a holomorphic map $$\phi:\frac{\mathbb{C}}{\Lambda}\to\frac{\mathbb{C}}{\Lambda'}$$

My aim is to lift it to a holomorphic map $\Phi:\mathbb{C}\to\mathbb{C}$. The author has hinted to use the lifting property for this.

I know that if $\tilde{X}$ is a covering space of $X$ then any map $f:Y\to X$ can be lifted to a map $\tilde{f}:Y\to\tilde{X}$. Since $\mathbb{C}$ is the covering space of any complex tori with covering map being the canonical projection map (correct me if I'm wrong). Hence my aim is to find a holomorphic $\Phi$ such that the diagram commutes.

$$\begin{array} A\mathbb{C} & \stackrel{\Phi}{\longrightarrow} & \mathbb{C} \\ \downarrow{p_1} & & \downarrow{p_2} \\ \frac{\mathbb{C}}{\Lambda} & \stackrel{\phi}{\longrightarrow} & \frac{\mathbb{C}}{\Lambda'} \end{array} $$ How do I proceed? Thank you.

Edit: I recently checked this question. How can we show that the lift of this holomorphic function will be holomorphic?

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The question you linked gives us the existence of the map you want. For it to be holomorphic it is enough to see that $p_1$ and $p_2$ are holomorphic and locally biholomorphic.

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  • $\begingroup$ But if $p_1$ is holomorphic and doubly periodic, it will turn out to be constant, right? $\endgroup$ – Swapnil Tripathi Nov 18 '16 at 14:13
  • $\begingroup$ You are defining the complex structure on the tori using the functions $p_1$ and $p_2$. They are just coordinates on the manifolds and the function $\phi$ is going to be holomorphic with respect to the coordinates $p_1$ and $p_2$. No $p_1$ and $p_2$ are not constants, liuville's theorem doesn't apply here. $\endgroup$ – hjr Nov 18 '16 at 14:21
  • $\begingroup$ I am having a major crisis in my head (I tried visualizing the map which seems wrong now). How do I proceed to show that $p_1$ is holomorphic? Maybe that'll give me some ideas. $\endgroup$ – Swapnil Tripathi Nov 18 '16 at 22:35
  • $\begingroup$ So what you need to show that $p_1$ is holomorphic is that for any point $z\in\mathbb{C}$ then there are open sets $U,V$. $U$ a neighborhood of $z$, $V$ a neighborhood of $p_1(z)$ and a chart $\psi:V\to W$ ($W$ a subset of $\C$ and $\psi$ a homeomorphism). Then $p_1$ is holomorphic at $z$ if $\psi\circ\p_1|U:U\to W$ is holomorphic. $\endgroup$ – hjr Nov 19 '16 at 5:17
  • $\begingroup$ Pressed enter too quickly, dont know how to edit my comment. But anyway, $\endgroup$ – hjr Nov 19 '16 at 5:18

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