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84 3. Elliptic Curves and Their Isomorphisms

Now we come to the construction of a family of elliptic curves, where each iso-
morphism class of curves is represented exactly once in the family. It begins with
a pair of coefficients. Although the concept of the moduli space of elliptic curves
is more involved in that automorphisms enter the picture, these next examples are
crude moduli spaces which are useful for explicit computations.

(8.5) Notation. In terms of equivalence classes of coefficients we introduce the fol-
lowing sets:
(1) isomorphism classes of elliptic curves E α, β :
2 3 2 ∗
E (K) = (K −{α = β })/K and
(2) isomorphism classes of possible singular elliptic curves with at most a double
2
∗

point E α, β : E (K) = (K −{(0, 0)})/K .
4
2
6
∗
In both cases the action of K on K is λ · (α, β) = (λ α, λ β).The J-function

is a bijection J : E (K) → K which extends to J : E (K) → P 1 (K) = K ∪
2
3
{∞}. The value J =∞ corresponds to the curve with a double point E λ ,λ at
(λ, 0) and third root giving the point (−2λ, 0). Hence sets of isomorphisms classes
2
of elliptic curves are parametrized by quotients of subsets of K .
2
(8.6) A Family on the Diagonal Subset of K . On the diagonal of all (α, α) ∈
2
2
3
(K −{(α, β) : α = β }) the J-function has the form
α J
J = J(α, α) = and solving for α it is α = .
α − 1 J − 1
This means that the equation of the curve has the following form, but with J = 1
excluded, that is, J ∈ K −{0, 1}
3J 2J J
2 3
E α, α : y = x − x + where α = β = .
J − 1 J − 1 J − 1
The family of curves E is a subset of (K −{0, 1})×P 2 (K) where (J : w : x : y) ∈ E
if and only if it satisfies the equation
J J

E ,
J − 1 J − 1
in homogeneous form
3J 2J
2 3 2 3
wy = x − w x + w .
J − 1 J − 1
The family has a projection π : E → (K −{0, 1}) defined by the restriction of the
product projection on the first factor, that is, π(J; w : x : y) = J.The fiber π −1 (J)
is the elliptic curve

J J
E , ,
J − 1 J − 1
and it is the unique curve in the family with this J-value for J ∈ (K −{0, 1}).