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§7. Real and Complex Points on Elliptic Curves 19

§7. Real and Complex Points on Elliptic Curves

Let E be an elliptic curve over the real or complex numbers. The structure of the
groups E(R) and E(C) as continuous groups or Lie groups is completely understood
with a little background on the subject of Lie groups. Using E for either the real or
the complex points of E, we point out several properties of these groups which allow
us to determine their structure from a general result.
(1) There is a topology (or notion of convergence) on E such that E is locally Eu-
clidean of dimension 1 in the real case and dimension 2 in the complex case.
This locally Euclidean property comes from the implicit function theorem since
E is nonsingular.
(2) The group operations are continuous, in fact, they are algebraic.
(3) The group E is a closed subspace of the projective plane and, since the projective
plane is compact, the group E is compact (every sequence has a convergent
subsequence).
Lie groups, which can be taken as locally Euclidean groups, have the following
structure under suitable assumptions.
(7.1) Assertion. An abelian, compact, and connected Lie group is isomorphic to a
product of circles. The number of factors is equal to the dimension of the locally
Euclidean space.
To check whether or not this applies to E, we graph E to check its connectivity.
2
Consider the elliptic curve given by the equation in normal form y + a 1 xy + a 3 y =
3
f (x), where f (x) = x + ··· is a cubic poynomial. Completing the square
2
a 1 x + a 3
∗
y + = f (x),
2
3
where f (x) = x +··· is also a cubic polynomial. Hence the graph of this equation
∗
for real coefficients is symmetric around the line 2y + a 1 x + a 3 = 0, so that is has
one of the following two forms:

In the case of one real root, the group E(R) has one connected component, and in
the case of three real roots, the group E(R) has two connected components. From
this observation and (7.1), we deduce the following result.