The j-invariant classifies complex tori

If ${\mathrm{\Lambda }}^{\mathrm{\prime }}=c\mathrm{\Lambda }$ then multiplication by $c$ is a biholomorphism $T_{\mathrm{\Lambda }}\to T_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}$ by [F2], so (i) implies (ii). Writing $\mathrm{\Lambda }=\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2={\omega }_1{\mathrm{\Lambda }}_{\tau }$ with $\tau ={\omega }_2\mathrm{/}{\omega }_1$, the lattice ${\mathrm{\Lambda }}^{\mathrm{\prime }}$ has the same normalised parameter $\tau$, so $j({\mathrm{\Lambda }}^{\mathrm{\prime }})=j(\tau )=j(\mathrm{\Lambda })$; hence (i) implies (iii). If ${\mathrm{\Lambda }}^{\mathrm{\prime }\mathrm{\prime }}$ is another basis of $\mathrm{\Lambda }$, it is related to $({\omega }_1,{\omega }_2)$ by a matrix in $S{L}_2(\mathbb{Z})$ and the new parameter is $\gamma \tau$, so $j$ is unchanged by [F1].

Suppose (ii): let $F:T_{\mathrm{\Lambda }}\to T_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}$ be a biholomorphism. Composing with a translation of $T_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}$ if necessary we may assume $F([0])=[0]$, since translations are biholomorphisms. Put $G:=F\circ \pi :\mathbb{C}\to T_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}$; as $\mathbb{C}$ is simply connected and ${\pi }^{\mathrm{\prime }}$ is a holomorphic covering [F3], there is a unique based lift $f:\mathbb{C}\to \mathbb{C}$ with ${\pi }^{\mathrm{\prime }}\circ f=G$ and $f(0)=0$. Similarly the inverse $F^{-1}$ admits a based lift $\tilde{f}$ with $\tilde{f}(0)=0$. Both $\tilde{f}\circ f$ and the identity are based lifts of $F^{-1}\circ F\circ \pi =\pi =\pi \circ \mathrm{id}$, so by uniqueness $\tilde{f}\circ f=\mathrm{id}$, and symmetrically $f\circ \tilde{f}=\mathrm{id}$; thus $f$ is a biholomorphism of $\mathbb{C}$ (it is holomorphic because locally it is a branch of the holomorphic covering ${\pi }^{\mathrm{\prime }}$ composed with $G$). By [F4] $f(z)=az+b$ with $a\mathrm{\ne }0$, and $f(0)=0$ gives $b=0$, so $f(z)=az$. For every $\lambda \in \mathrm{\Lambda }$, ${\pi }^{\mathrm{\prime }}(f(z+\lambda ))=G(z+\lambda )=G(z)$ because $\pi$ is $\mathrm{\Lambda }$-periodic, so $f(z+\lambda )-f(z)=a\lambda \in \ker {\pi }^{\mathrm{\prime }}={\mathrm{\Lambda }}^{\mathrm{\prime }}$; hence $a\mathrm{\Lambda }\subseteq {\mathrm{\Lambda }}^{\mathrm{\prime }}$. Applying the same argument to $\tilde{f}$, whose affine form is $a^{-1}z$, gives $a^{-1}{\mathrm{\Lambda }}^{\mathrm{\prime }}\subseteq \mathrm{\Lambda }$, so $a\mathrm{\Lambda }={\mathrm{\Lambda }}^{\mathrm{\prime }}$ and ${\mathrm{\Lambda }}^{\mathrm{\prime }}$ is homothetic to $\mathrm{\Lambda }$: (ii) implies (i).

Suppose (iii): $j(\tau )=j({\tau }^{\mathrm{\prime }})$ where $\mathrm{\Lambda }={\omega }_1{\mathrm{\Lambda }}_{\tau }$, ${\mathrm{\Lambda }}^{\mathrm{\prime }}={\omega }_1^{\mathrm{\prime }}{\mathrm{\Lambda }}_{{\tau }^{\mathrm{\prime }}}$; set $\mu =j(\tau )=j({\tau }^{\mathrm{\prime }})$. The form $f:=E_4^3-\mu \mathrm{\Delta }\in M_{12}$ has $f(\tau )=0=f({\tau }^{\mathrm{\prime }})$ and $f(\mathrm{\infty })=1$, so $f\mathrm{\ne }0$ and by [F5] all zeros of $f$, in particular $\tau$ and ${\tau }^{\mathrm{\prime }}$, lie in one $PS{L}_2(\mathbb{Z})$-class: there is $\gamma \in S{L}_2(\mathbb{Z})$ with ${\tau }^{\mathrm{\prime }}=\gamma \tau$. Then ${\mathrm{\Lambda }}_{{\tau }^{\mathrm{\prime }}}={\mathrm{\Lambda }}_{\gamma \tau }=(c\tau +d{)}^{-1}{\mathrm{\Lambda }}_{\tau }$ by [F6] is homothetic to ${\mathrm{\Lambda }}_{\tau }$, hence ${\mathrm{\Lambda }}^{\mathrm{\prime }}$ is homothetic to $\mathrm{\Lambda }$ and (i) holds. The three implications close the equivalence.