The elliptic curves (or complex tori) can be parametrized in 2 different ways. The first method parametrizes lattices in the complex plane in a rather obvious way. The second parametrization gives to each elliptic curve a more geometric value in the sense that this value corresponds more closely to how the curve is embedded in the plane. Then there is a function mapping the first parametrization to the other. This function is called the $j$-invariant and it is a modular form of weight zero, where number theory comes to join geometry and algebra. We will discuss briefly what modular forms are and what a fundamental domain is--all the absolute basics you need to know.