El teorema de Nagell-Lutz

Abstract

The goal of this memory is to prove the Nagell-Lutz theorem, an important theorem about the rational torsion of elliptic curves, which gives us a necessary condition for a point to have finite order. To do this, we will start by studying the geometry of elliptic curves, how to endow them with an abelian group structure, the group formulas and some birational transformations that put the elliptic curve into the so-called Weierstrass normal form. Given an elliptic curve in Weierstrass normal form, we will show that if a point has finite order, then its coordinates must be integers, and we will summarize all the work in the proof of the theorem.