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§7. Singular Cubic Curves 81

2
2
1 − a 1 t − a 2 t = 0. The discriminant of this quadratic is b 2 = a + 4a 2 .Infact, we
1
can go further and put a group structure on C ns using the chord-tangent construction
considered in Chapter 1. Unlike the nonsingular cubics which give entirely new ob-
jects of study, singular cubics are isomorphic to the familiar multiplicative G m and
additive G a groups. This relation is carried out explicitly in the next theorem.
3
2
(7.2) Theorem. Let E be a cubic curve over k with equation y +a 1 xy = x +a 2 x 2
3
which we factor as (y − αx)(y − βx) = x over the field k 1 = k(α) = k(β).
(1) Multiplicative case, α = β: The function (x, y) → (y − βx)/(y − αx) defines
a homomorphism E ns → G m over k 1 .
∗
(a) If k = k 1 , that is, α and β are in k, then the mapE ns (k) → G m (k) = k is
an isomorphism onto the multiplicative group of k.
(b) If k 1 is a quadratic extension of k, that is, α and β are not in k, then the map
∗
∗
defines an isomorphism E ns (k) → ker(N k 1 /k ), where N k 1 /k : k → k is
1
∗
the norm mapand ker(N k 1 /k ) is the subgroupelements in k with norm 1.
1
(2) Additive case, α = β: The function (x, y) → x/(y − αx) defines a homomor-
phism E ns → G a over k 1 . The mapE ns (k 1 ) → G a (k 1 ) is an isomorphism onto
the additive groupof k 1 . Observe that k = k(α) except possibly in characteristic
2.
Proof. In the multiplicative case we introduce the new variables
y − βx 1
u = and v = .
y − αx y − αx
3
3
3
3 3
Using the relation (y −αx) (u −1) = (α −β) x = (α −β) (y −αx)(y −βx),we
3
3
obtain the equation for E ns in (u,v)-coordinates as (α−β) uv = (u−1) . Moreover,
lines in x, y with equations Ax + By +C = 0 are transformed into lines in u,v with

equations A u + B v + C = 0. If (u 1 ,v 1 ), (u 2 ,v 2 ),and (u 3 ,v 3 ) are three points on
the cubic E ns which lie on a line v = λu + δ, then we have the factorization
3
3
0 = (u − 1) − (α − β) u(λu + δ) = (u − u 1 )(u − u 2 )(u − u 3 ),
and hence the relation u 1 u 2 u 3 = 1 in the multiplicative group. This means that the
function (x, y) → u carries the group law on E ns into the multiplicative group law
∗
on k .
1
Finally the elements u = (y − βx)/(y − αx) have norm one
y − βx y − αx

N k 1 /k (u) = uu = · = 1,
y − αx y − βx

where α = β and β = α are conjugates of each other in k 1 over k. Conversely, if

z in k 1 has norm 1, then for some c in k 1 the element w = c + zc = 0 and from

w = c + z c we deduce that

zw = zc + zz c = c + zc = w.