Is there any well written introduction for the modular space of complex structures on $T^2$?



Any introduction to Teichmuller Space of $T^2$?
As requested, I'm promoting this comment to an answer: 


McKean and Moll's book "Elliptic curves" is a basic introduction to 2-tori with complex structure from the function theoretic, geometric, and arithmetic perspectives. What's closest to what you want is discussed in section 2.6 (on moduli of elliptic curves) and chapter 4 (on the modular group).


