# For an elliptic curve E, does there exist a cofinite Fuchsian group without elliptic elements with quotient E minus a finite subset

Let $E$ be a compact Riemann surface of genus 1, i.e., an elliptic curve.

Let $P$ be the identity element of $E$.

Question 1. Does there exist a cofinite Fuchsian group (or a Fuchsian group of the first kind) $\Gamma$ such that $E-\{P\}=\Gamma\backslash\mathfrak{h}$, where $\mathfrak{h}$ denotes the complex upperhalf plane, and $E$ can be obtained by adding the ''cusp'' $P$?

Question 2. Does there exist a cofinite Fuchsian group $\Gamma$ without elliptic elements and a finite set of points $\{b_1,\ldots,b_n\}\subset E$ such that $E-\{b_1,\ldots,b_n\} =\Gamma\backslash\mathfrak{h}$ and $E$ can be obtained by adding the ''cusps'' $b_i$.

If yes, how are these groups related to $E$? What can be said about them?

I would like to choose $b_1,b_2,b_3$ and $b_4$ to be the ramification points of the projection onto the $x$-coordinate, i.e., Weierstrass P-function.

-
the answer to both question is yes, by the uniformization theorem (for any choice $b_1,\dots,b_n$ ($n>0$) it will give you a $\Gamma$ acting freely, i.e. without elliptic elements). – user8268 Jun 24 '11 at 20:42
And this Gamma will be a discrete subgroup of SL_2(R) of finite volume because....? – Narumi Jun 24 '11 at 22:48
It (i.e. finite volume) is true for any compact Riemann surface with finitely many points removed. The proof boils down to showing that a loop around any of the removed points corresponds to a parabolic transformation. – user8268 Jun 25 '11 at 19:40