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§6. Periods Associated with Elliptic Curves: Elliptic Integrals 183

§6. Periods Associated with Elliptic Curves: Elliptic Integrals

Let E be an elliptic curve defined over the complex numbers by an equation
2
y = f (x), where f (x) is a cubic equation in x. We wish to determine a complex
torus C/L with L = Zω 1 + Zω 2 such that (℘, ℘ ) : C/L → E(C) is an analytic

isomorphism. The lattice L is determined by generators ω 1 and ω 2 . Our goal is to
describe ω 1 and ω 2 in terms of E using the fact that these generators are examples
of periods of integrals
( (
ω 1 = θ and ω 2 = θ,
C 1 C 2
where C 1 and C 2 are suitably chosen closed curves on E(C) and
dx dx 1
θ = = = dz (x = ℘(z))

f (x) 2y 2
is the invariant differential on the cubic curve. In terms of the period parallelogram
these integrals are just the integral of dz over a side of the parallelogram, hence
a period of the lattice. The suitably chosen closed paths on E(C) are determined
by considering x(z) = ℘(z) mapping a period parallelogram onto the Riemann
sphere and looking at the four ramification points corresponding to the half periods
ω 1 /2,ω 2 /2,(ω 1 + ω 2 )/2, and 0.

Implicit in the above construction is a basis of the 2-division points (r 1 , 0), (r 2 , 0)
corresponding to ω 1 ,ω 2 . For explicit calculations we normalize the roots of f (x) by
transforming r 1 to 0, r 2 to 1, and r 3 to some λ. The equation for the elliptic curve is
2
in Legendre form E λ : y = x(x − 1)(x − λ) for λ ∈ C −{0, 1}. We know from
1(6.3) that there are six possible choices of λ, namely