Exercise 2, § 1

Let H be the set of complex numbers $\tau$ such that $\mathscr{I}(\tau )>0$.

a) Show that an analytic law of left operation of the discrete group $\mathbf{S}\mathbf{L}(2,\mathbf{Z})$ on H can be defined by putting $\gamma \tau =\frac{a\tau+b}{c\tau+d}$ for all $\gamma =(ab)\in \mathbf{S}\mathbf{L}(2,\mathbf{Z})$

and $\tau \in H$.

b) For $\tau \in H$, denote by $T_{\tau}$ the complex torus $\mathbf{C}/(\mathbf{Z}+\mathbf{Z}\tau )$. Show that the map $\tau \rightarrow T_{\tau}$ induces by passage to the quotient a bijective map from the set $H/\mathbf{S}\mathbf{L}(2,\mathbf{Z})$ to the set of isomorphism classes of complex tori of dimension one.