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This is something that I perhaps would be expected to know, but don't. Let $E_\tau$ be the elliptic curve ${\mathbb C}/({\mathbb Z +\mathbb Z} \tau)$. Consider the complement of a point in $E_\tau$, call it $E^0_\tau=E_\tau-\{{\mathbb Z +\mathbb Z} \tau\}$.

I believe that the universal cover of $E^0_\tau$ is biholomorphic to the upper half plane $H$. Is there any explicit way of seeing this universal cover $H\to E_\tau^0=H/G_\tau$ ?

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  • $\begingroup$ That the universal cover must be biholomorphic to $H$ follows easily from the uniformization theorem and Picard's theorem. Just note that a holomorphic map $\mathbb{C} \to \mathbb{C}/\Lambda_{\tau}$ factors through an entire function $\mathbb{C} \to \mathbb{C}$. By Picard, its range may not miss the infinite set $\mathbb{Z} + \mathbb{Z}\tau$ unless the map is constant. So there are no non-constant holomorphic mappings $\mathbb{C} \to E_{\tau}^0$, and the Riemann surface $E_{\tau}^0$ must be hyperbolic. $\endgroup$
    – Vesselin Dimitrov
    Commented Jul 22, 2014 at 22:43
  • $\begingroup$ Right. I was also thinking that there is no way of getting anything like this from $\mathbb C$ or $\mathbb C\mathbb P^1$. The trouble is that this is far from explicit. $\endgroup$
    – Lev Borisov
    Commented Jul 22, 2014 at 22:53
  • $\begingroup$ The fundamental group of $E_{\tau}^0$ is free on two generators. It should be not too difficult to write this fundamental group as a Fuchsian group $\Gamma \subset \mathrm{PSL}(2, \; \mathbb{R})$, and to find a (necessarily non-compact) fundamental region for $\Gamma$ in $\mathbb{H}$. $\endgroup$
    – Francesco Polizzi
    Commented Jul 22, 2014 at 22:58
  • $\begingroup$ Indeed, the commutator becomes the loop around the puncture. Unfortunately, I don't know much about Fuchsian groups. It seems that the formulas should involve $\tau$, but how? $\endgroup$
    – Lev Borisov
    Commented Jul 22, 2014 at 23:02
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    $\begingroup$ A covering map $\Delta \to \mathbb{C} - (\mathbb{Z} + \mathbb{Z}\tau)$ can be constructed by the Schwarz reflection principle. mathoverflow.net/questions/141372/… ; arxiv.org/pdf/1110.2696.pdf $\endgroup$
    – Vesselin Dimitrov
    Commented Jul 22, 2014 at 23:14

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