Corbes el·líptiques, superfícies de Riemann i tors complexos

Abstract

[en] Elliptic curves are a mathematical object that has been studied by mathematicians since Diophantus. Even though in some cases it was in disguise, a lot of problems in number theory, analisis or geometry presented by an enormous number of mathematicians involved elliptic curves. Some names that can not be overlooked are Diophantus, Fibonacci, Fermat, Euler, Newton, Jacobi, Weierstrass or Poincaré (even though there are a lot of other names worth mentioning). Riemann surfaces are yet another object of great interest that can be observed from a lot of different perspectives. When they were first introduced by Riemann, Klein and Weyl, it was made visible that they could be observed as one-dimensional complex manifolds or algebraic curves. Although there has been other interpretations (for exemple to see them as two-dimensional real manifolds), we will center around these first two. If we observe a Riemann surface as an algebraic curve, an interesting question arises; is there any relation between Riemann surfaces and elliptic curves? The purpose of this project is to see that the answer is yes. Furthermore, we will see that an elliptic curve is equivalent to some kind of Riemann Surfaces and its properties.