Different complex structures of a torus. Consider a complex $\mathbb C/\Lambda$, once we choose a lattice $\Lambda$, then the torus is uniquely determined.
For example, if we choose different lattices: $\Lambda_1=\mathbb Z\oplus i\mathbb Z$, $\Lambda_2=\mathbb Z\oplus 2i\mathbb Z$, and let $T_1=\mathbb C/\Lambda_1$, $T_2=\mathbb C/\Lambda_2$, then do $T_1$ and $T_2$ have different complex structures? I know they have the same diffeomorphism sturcture, but why they have different complex structures? Does there exist an intuitive explanation why they have different complex structures?
 A: Here is a general procedure used to determine if two tori are biholomorphic to each other. Consider a lattice $\Lambda$ in ${\mathbb C}$ generated by two complex numbers $\alpha, \beta$. I assume that the basis $(\alpha, \beta)$ in ${\mathbb R}^2$ is positively oriented, otherwise, swap $\alpha$ and $\beta$. Consider the ratio $\tau=\beta/\alpha$ (the Teichmuller parameter). Positive orientation implies that $\tau$ is in the upper half-plane $U$, $Im(\tau)>0$. Then, find an element $\gamma\in PSL(2, {\mathbb Z})$ sending $\tau$ to the modular fundamental domain
$$
F=\{z: - 1/2 \le Re(z)\le 1/2, |z|\ge 1\}. 
$$
If $z=\gamma(\tau)$ is in the interior of $F$, then it is uniquely determined by $\tau$, otherwise, it is unique up to the action of the translation $z\mapsto z\pm 1$ (if $Re(z)=\pm 1/2$) or involution $z\mapsto -1/z$ (if $|z|=1$).
Up to this ambiguity, $z$ uniquely determines the conformal class of the quotient torus ${\mathbb C}/\Lambda$.
In the case of a "rectangular" torus, things are especially simple since $Re(\tau)=0$ and either $\tau\in F$ or $-1/\tau\in F$.
Now, to your specific example (from your comment): $\Lambda_2$ is generated by $\alpha=2$ and $\beta=i$. In this case, $\tau=i/2$, $-1/\tau=2i\in F$. Hence, the torus ${\mathbb C}/\Lambda_2$ is not conformal to the torus ${\mathbb C}/\Lambda_1$.

Regarding references: This Wikipedia article provides a nice proof of the fact that the moduli space of complex one-dimensional tori (aka nonsingular complex elliptic curves) is the quotient of the upper half-plane $U$ by  $\Gamma=PSL(2, {\mathbb Z})$. Concretely, this means that if $\Lambda_1, \Lambda_2$ are two lattices generated, respectively, by $\alpha_1, \beta_1$ and $\alpha_2, \beta_2$ with $\tau_k=\beta_k/\alpha_k, k=1, 2$, then the tori ${\mathbb C}/\Lambda_k$ are biholomorpbhic if and only if
$$
\Gamma \tau_1 =\Gamma \tau_2. 
$$
The fact that this equality of two orbits is equivalent to
$$
\Gamma \tau_1 \cap F= \Gamma \tau_2 \cap F,
$$
i.e. that $F$ is a fundamental domain of $\Gamma$, is explained in many places. One reference is given by the linked Wikipedia article, Chapter VII in
Serre, Jean-Pierre, A course in arithmetic, Graduate Texts in Mathematics. 7. New York-Heidelberg-Berlin: Springer-Verlag. viii, 115 p. (1973). ZBL0256.12001.
Another is
Katok, Svetlana, Fuchsian groups, Chicago Lectures in Mathematics. Chicago: The University of Chicago Press,. x, 175 p. (1992). ZBL0753.30001.
Or, freely available Pete Clark's lecture notes on Shimura curves here.
Lastly, a true story: A math professor is flying in an airplane, reading Serre's book "A course in arithmetic." His next-seat neighbor notices the title and approvingly says "Very good, it is never too late to learn!"
