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:

1The

contents of this chapter are largely based on the article [Loz05], in Spanish.

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http://dx.doi.org/10.1090/stml/058/02