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§8. Parameterization of Curves in Characteristic Unequal to 2 or 3 83

(8.2) Nonsingular Curves. E α, β . We study the question of when E α, β is non-
2
3
singular using (d/dx)(x − 3αx + 2β) = 3x − 3α. Multiple roots of the cubic are
solutions of the two equations
3
3
0 = x − 3αx + 2β and 0 = 3x − 3αx.
3
Eliminating the x term, we obtain −6αx + 6β = 0or x = β/α as the only possible
2
2
multiple root. Substituting this back into the multiple α (3x −3α) of the derivative,
2 3
we obtain 3(β − α ), and hence E α, β is nonsingular if and only if (α, β) =
2
3
= α − β = 0.
(8.3) Isomorphism Classification. We introduce the invariant
α 3 α 3
J(α, β) = =
3
α − β 2 (α, β)
for E α, β . Then the isomorphism classification given in (4.3) takes the form of the
following three equivalent statements:

(1) the elliptic curves E α, β and E α ,β are isomorphic,

4
6

∗
(2) there exists λ ∈ K with λ α = α and λ β = β ,and
(3) the J-invariants are equal J(α, β) = J(α ,β ).

To construct a space with one point representing each isomorphism class of an
elliptic curve over K we use the following action.
2
(8.4) A Filtration of K Stable under the Action of the Multiplicative Group.
6
2
4
∗
The multiplicative group K of K acts on K by the formula λ·(α, β) = (λ α, λ β),
and J is equivariant by the above (8.3) (2) and (3), that is, J(λ · (α, β)) =
6
2
4
∗
J(λ α, λ β) = J(α, β). We exhibit a K -equivariant filtration of K which maps
by J to a filtration of the projective line P 1 (K) under J as follows
2
3
2
2
∗ 2
2
3
K 2 ⊃ K −{(0, 0)}⊃ K −{α = β }⊃ (K ) −{α = β }
  
  
J J J
P 1 (K) ⊃ K ⊃ K −{0, 1}.
Now (8.3) can be summarized in terms of this filtration as follows:
(1) E 0, 0 is the cuspidal singular cubic.
2
3
(2) J(α, β) =∞ if and only if α = 0and α = β , in which case E α, β is
singular cubic with a double point at the origin.
2
3
(3) J(α, β) = 0 if and only if the curve is E 0,β : y = x + 2β, β = 0, with an
automorphism group of order 6.
2
3
(4) J(α, β) = 1 if and only if the curve is E α, 0 : y = x − 3αx,α = 0, with
automorphism group cyclic of order 4.