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
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