20 2. Elliptic curves

then C(C) may be considered as a surface over R, and the

genus of C counts the number of holes in the surface. For

example, the projective line

P1(C)

has no holes and g = 0

(the projective plane is homeomorphic to a sphere; see Ap-

pendix C for a quick introduction to projective geometry),

and an elliptic curve has genus 1 (homeomorphic to a torus;

see Theorem 3.2.5). Surprisingly, the genus of a curve is

intimately related with the arithmetic of its points. More

precisely, Louis Mordell conjectured that a curve C of genus

≥ 2 can only have a finite number of rational solutions. The

conjecture was proved by Faltings in 1983.

2.2. Definition

Definition 2.2.1. An elliptic curve over Q is a smooth cubic projec-

tive curve E defined over Q with at least one rational point O ∈ E(Q)

that we call the origin.

In other words, an elliptic curve is a curve E in the projective

plane (see Appendix C) given by a cubic polynomial F (X, Y, Z) = 0

with rational coeﬃcients, i.e.,

F (X, Y, Z) =

aX3

+

bX2Y

+ cXY

2

+ dY

3

(2.2)

+eX2Z

+ fXY Z + gY

2Z

+hXZ2

+ jY

Z2

+

kZ3

= 0,

with coeﬃcients a, b, c, . . . ∈ Q, and such that E is smooth; i.e., the

tangent vector

(

∂F

∂X

(P),

∂F

∂Y

(P),

∂F

∂Z

(P)

)

does not vanish at any P ∈ E

(see Appendix C.5 for a brief introduction to singularities and non-

singular or smooth curves). If the coeﬃcients a, b, c, . . . are in a field

K, then we say that E is defined over K (and write E/K).

Even though the fact that E is a projective curve is crucial, we

usually consider just aﬃne charts of E, e.g. those points of the form

{[X, Y, 1]}, and study instead the aﬃne curve given by

aX3

+

bX2Y

+ cXY

2

+ dY

3

(2.3)

+eX2

+ fXY + gY

2

+ hX + jY + k = 0