In fact, for the cases \#1,\#2,\#3, \#4 and \#5, the lifts are the same and respectively given by: $\Gamma(4)$, $\Gamma_0(3)\cap\Gamma(2)$,$\Gamma_1(7)$,$\Gamma_1(8)$ and $\Gamma_0(8)\cap\Gamma(2)$. In the cases \#6,\#7,\#8 and \#9, the lifts are not unique, and we choose respectively the following lifts: $\Gamma_1^{\prime}(8;4,1,2)$, $\Gamma_0(12)\cap\Gamma_1(3)$, $\Gamma_0(16)\cap\Gamma_1(4)$ and $\Gamma_1^{\prime}(16;16,2,2)$. Here $\Gamma_1^{\prime}(8;4,1,2)$ and $\Gamma_1^{\prime}(16;16,2,2)$ are defined in the same way as $\Gamma_1(8;4,1,2)$ and $\Gamma_1(16;16,2,2)$ but without the $\pm$. Note that the widths of the cusps are not affected by taking a lift as $-{\rm Id}$ is the only difference.

(We should remark that the lifts are not unique; for instacne, in the case \#7, there are four lifts, but only one has no elements of trace $-2$, which is the one given above.)

In fact, Proposition 4 has also been obtained by A. Sebbar in his unpublished note.

Proposition 4 will pave a way to the definition of elliptic modular surfaces, which we will discuss next.

[**Elliptic Modular Surfaces**]{}: In \[Sh1\] Shioda has shown how to associate to any subgroup $G$ of $SL(2,{{\mathbb Z}})$ of finite index and not containing $-$Id an elliptic fibration $E(G)$, called the elliptic modular surface associated to $G$, over the modular curve $X(G)=\overline{G\backslash{\frak{H}}}$.

(Shioda’s construction requires that modular groups ought 