An elliptic curve over the field of complex numbers $\mathbb{C}$ is a nonsingular projective cubic, given by the Weierstrass equation. Using the implicit function theorem we show, that it admits a complex structure, which makes it a Riemann surface. On the other hand every torus, which we view as a quotient of $\mathbb{C}$ modulo a lattice $\Lambda$ - a discrete subgroup of $\mathbb{C}$ isomorphic to $\mathbb{Z}^2$, inherits a complex structure via its quotient map, which justifies us naming it complex torus. These two, vastly different constructions, remarkably turn out to yield the same i.e. isomorphic mathematical objects. This link is explained using the theory of doubly periodic functions, also called elliptic functions, and modular functions, which are meromorphic functions on the upper half plane enjoying many symmetries involving the modular group $\operatorname{SL}_2(\mathbb{Z})$. We discuss their basic properties and develop the necessary theory of the Weierstrass $\wp$-function, alowing us to explicitly construct the aforementioned isomorphism and prove the uniformization theorem, unifying both objects of study.
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