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§5. Remarks on the Group Law on Singular Cubics 43

and 0 at infinity in the extended x, y-plane transforms to 1. If (u 1 ,v 1 ), (u 2 ,v 2 ),and
(u 3 ,v 3 ) are three points on a line v = λu + δ, then
3
0 = (u − 1) − 8u(λu + δ) = (u − u 1 )(u − u 2 )(u − u 3 ),
and thus we have u 1 u 2 u 3 = 1. This means that (x, y) → (y + x)/(y − x) is a group
3
2
homomorphism of the nonsingular points of the cubic curve E given by y = x +x 2
∗
into the multiplicative group k = k −{0}.
2
2
Let E be the cubic curve y = x (x +1) together with 0 at infinity, and let E ns (k)
3
denote E(k) −{0}. Given a nonzero u we define v by the equation 8uv = (u − 1) .
From
1 u
y − x = and y + x = ,
v v
we have the definition of
u − 1 u + 1
x = and y = .
2v 2v
These formulas require that 2 is nonzero in k, and with the above discussion yield
the following proposition.
(5.4) Proposition. With the above notations for E over a field k of characteristic
∗
= 2 the function f : E ns (k) → k given by f (0) = 1 and f (x, y) = (y − x)(y + x)
is a group isomorphism of the k-valued points on E ns onto the multiplicative group
of k.

Exercise

1. Show that (0, 0) is a singular point on
2 3 2
y + a 1 xy = x + a 2 x
by change of coordinates. Determine conditions on a 1 and a 2 when it is a double point
andwhenitisacusp.