Chapter 2 Elliptic curves In this chapter we summarize the main aspects of the theory of el- liptic curves1. Unfortunately, we will not be able to provide many of the proofs because they are beyond the scope of this course. If the reader is not familiar with projective geometry or needs to refresh the memory, it is a good time to look at Appendix C or another reference (for example, [SK52] is a beautiful book on projective geometry). 2.1. Why elliptic curves? A Diophantine equation is an equation given by a polynomial with integer coeﬃcients, i.e. f(x1,x2,...,xr) = 0 (2.1) with f(x1,...,xr) ∈ Z[x1, . . . , xr]. Since antiquity, many mathe- maticians have studied the solutions in integers of Diophantine equa- tions that arise from a variety of problems in number theory, e.g. y2 = x3 − n2x is the Diophantine equation related to the study of the congruent number problem (see Example 1.1.2). Since we would like to systematically study the integer solutions of Diophantine equations, we ask ourselves three basic questions: 1 The contents of this chapter are largely based on the article [Loz05], in Spanish. 17 http://dx.doi.org/10.1090/stml/058/02

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.