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20 Introduction to Rational Points on Plane Curves

(7.2) Proposition. Let E be an elliptic curve defined by an equation in the form
2
(y + ax + b) = g(x), where g(x) is a cubic polynomial over the real numbers.
If g(x) has only one real root, then E(R) is isomorphic as a Lie group to a circle,
and if g(x) has three real roots, then E(R) is isomorphic to a circle direct sum with
Z/2Z.

Over the complex numbers every elliptic curve is connected so that the corre-
sponding situation is easier to describe.

(7.3) Proposition. Let E be an elliptic curve defined over the complex numbers.
Then E(C) is isomorphic as a Lie group to the product of two circles, hence, it is a
torus.

In the chapter on elliptic functions we will give a proof of this result using
complex analysis and an explicit mapping using elliptic functions. Moreover, E(C)
will appear as C/L, where L is a lattice in the plane having a basis of two ele-
ments. In other words E(C) is isomorphic to R/Z × R/Z, as asserted above, and
L = Zω 1 + Zω 2 with Im (ω 2 /ω 1 ) = 0.
(7.4) Remark. From this we see that the kernel of multiplication by n is isomor-
phic to Z/nZ × Z/nZ. By contrast the finite subgroups of E(R) are of the form a
cyclic group or a cyclic group direct sum with the group of order 2, i.e., of the form
Z/nZ or Z/nZ × Z/2Z up to isomorphism. Since for an elliptic curve E over the
rational numbers E(Q) ⊂ E(R), the same holds for finite subgroups of E(Q),and
in particular for the torsion subgroup Tors E(Q) of E(Q) as remarked in (5.3).
Finally there is the question of why projective space over the real numbers or over
the complex numbers is compact. This follows because they are separated quotient
spaces of spheres.

(7.5) Remark. Each point in the real or complex projective plane has homogeneous
2
2
2
coordinates w : x : y where |w| +|x| +|y| = 1. The real projective plane P 2 (R)
2
3
is a quotient of the 2-sphere S in R where (w, x, y) and (w , x , y ) give the same

point in P 2 (R) if and only if w = uw, x = ux,and y = uy, where u =±1.

3
5
The complex projective plane P 2 (C) is a quotient of the 5-sphere S in C where

(w, x, y) and (w , x , y ) give the same point in P 2 (C) if and only if w = uw, x =
ux,and y = uy, where |u|= 1.

§8. The Elliptic Curve Group Law on the Intersection of Two
Quadrics in Projective Three Space
The content of this section is only sketched and not used in the rest of the book.
Supplying the details is a serious exercise.
In Sections 4 and 5 we introduced elliptic curves as certain cubic curves in the
projective plane, and using the intersection properties of lines and cubics, we defined