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82 3. Elliptic Curves and Their Isomorphisms

Hence z = w/w = (y − βx)/(y − αx) for x, y in k.
In the additive case we introduce the new variables
x 1
u = and v = .
y − αx y − αx
2 3 3 3
Using the relation (y − αx) = x = (y − αx) u , we obtain the equation for E ns
3
in (u,v)-coordinates as v = u . Moreover, lines in x, y are transformed into lines in
u,v.If (u 1 ,v 1 ), (u 2 ,v 2 ),and (u 3 ,v 3 ) are three points on the cubic E ns which lie of
a line v = λu + δ, then we have the factorization
3
0 = u − (λu + δ) = (u − u 1 )(u − u 2 )(u − u 3 ),
and hence the relation u 1 + u 2 + u 3 = 0 in the additive group. This means that the
function (x, y) → u carries the group law on E ns into the additive group law on k 1 ,
+
and E ns (k 1 ) → k is an isomorphism of groups. This proves the theorem.
1
(7.3) Remark. At the singular point (0, 0) on the cubic
3
2
x = y + a 1 xy − a 2 x 2
= (y − αx)(y − βx)
the tangent lines are given by y = αx and y = βx. The discriminant of the quadratic
2
form factoring into the equations of the tangent lines is D = a +4a 2 = b 2 . The two
1
cases considered in the previous theorem correspond to two kinds of singularities
and significantly for further questions to two kinds of j values:
(1) The singularity (0, 0) is a node (for simple double point) if and only if D =
b 2 = 0, i.e., α = β. Observe that β 2 = 0, c 4 = 0, and c 6 = 0 are all equivalent
in this case and
c 3
j = 4 =∞, where = 0.

∗ 2
The tangents are rational over k if and only if b 2 is a square in k, i.e., b 2 ∈ (k ) .
(2) The singularity (0, 0) is a cusp if and only if D = b 2 = 0, i.e., α = β.Observe
that β 2 = 0, c 4 = 0, and c 6 = 0 are all equivalent in this case and j = 0/0is
indeterminate.

§8. Parameterization of Curves in Characteristic Unequal to
2or3

In this section K is a field of characteristic = 2, 3.
(8.1) Notation. For α, β ∈ K we denote the elliptic curve with cubic equation
2 3
y = x − 3αx + 2β
by E α, β .