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§4. Elliptic Curves over the Real Numbers 285

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(4.1) Notation. Let K be a field, and consider the curve E"α, β#: y = x − 3αx +

2β.Wehaveseenthat E"α, β# and E"α ,β # are isomorphic if and only if there
6
4

exists λ ∈ K with λ α = α and λ β = β . Since the J-value of E"α, β# is

∗
3
2
3
J(α, β) = α / (α, β) where (α, β) = α − β , we introduce the following two
orbit spaces
2
3
2
2

∗
E (K) = K −{α = β }/K ∗ and E (K) = K −{(0, 0)}/K .
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2
4
∗
In both cases the action of K on K is by λ · (α, β) = (λ α, λ β).Wehavecon-
2
3
2
∗
structed a bijection from E (K) = K −{α = β }/K to isomorphism classes of
2

elliptic curves and a bijection from E (K) = K −{(0, 0)}/K to isomorphism
∗
classes of possibly singular elliptic curves with at most a double point.
For K algebraically closed the J-function is a bijection J : E (K) → K which

extends to a bijection J : E (K) → K ∪{∞} where P 1 (K) = K ∪{∞}. The value
3
2
J =∞ corresponds to the curve with a double point E"λ ,λ # at λ and third root at
−2λ.
(4.2) The Upper Half Plane and the Cubic Curve. Recall the bijection
: H/SL(2, Z) → E (C)
where τ ∈ H is assigned to the curve (τ) = E"E 4 (τ), E 6 (τ)# with cubic equation
2
3
y = x − 3E 4 (τ)x + 2E 6 (τ),
see 10(1.6). We diagram both the smooth and the singular cubics for both C and R

3
Recall the classical j = 12 J. Now we study which points in H/SL(2, Z), or equiv-
alently, in the fundamental domain of H, map under to E (R) ⊂ E (C). Then
we describe the behavior of J on these domains.

(4.3) Eisenstein Series Coefficients. The cubic equation in the Weierstrass theory
was originally derived in the form,

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3
y = 4x − g 2 (τ)x − g 3 (τ),