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Elliptic Curves and Their Isomorphisms

Using the results of the previous chapter, we complete the unfinished business con-
cerning elliptic curves described as cubic curves, namely the associative law, the
transformation into normal form, and the discriminant criterion for nonsingularity or
smoothness. At the same time admissible changes of variables are introduced; these
are equivalent to isomorphisms given by a change of variable in the first equation
from an elliptic curve defined by one cubic onto another.
The special cases of characteristic 2 and 3 are considered in detail. All elliptic
curves over F 2 , the field of two elements, are described and their isomorphism re-
lations over extension fields are given. Finally, we return to the subject of singular
cubics and their group of nonsingular points.

§1. The Group Law on a Nonsingular Cubic

In the Introduction we described the chord-tangent group law on a nonsingular cubic
curve, and in Chapter 1 we made extensive calculations with this group law. Now
using the intersection theory of Chapter 2, we show that the group law satisfies the
associative law and point out how the intersection multiplicity i(P, L, C) enters into
the definition of the group law.
For the chord-tangent composition PQ of P and Q on a nonsingular cubic the
following assertions based on 2(3.1), 2(4.5), and 2(4.6) are used.
(1.1) Remarks. Let L be alineand C a cubic curve both defined over a field k.Let
k be an algebraically closed extension of k. One of the following situations hold for

the intersection L(k ) ∩ C(k ):

(a) L(k ) ∩ C(k ) ={P 1 , P 2 , P 3 }, three points where i(P i , L, C) = 1 for i =
1, 2, 3. The composition is given by P i P j = P k ,and if P i and P j are rational over
k,thensois P k for {i, j, k}={1, 2, 3}.

(b) L(k )∩C(k ) ={P, P }, two points where i(P; L, C) = 2and i(P ; L, C) =
1. Either L is tangent to C at P or P is a singular point of C, and the compositions

are given by PP = P and PP = P.If P is rational over k,thensois P .