Reference for transformation period integrals into elliptic integrals

Question

The theory of elliptic functions tells us that any elliptic curve defined by a cubic $y^2 = 4(x - e_1(x - e_2)(x - e_3)$ with distinct roots is isomorphic to the quotient $\mathbb{C}$ by a lattice. The periods of this lattice can be found by evaluating the period integrals $$\int_{\infty}^{e_1}[(x - e_1)(x - e_2)(x - e_3)]^{-1/2}dx$$ and $$\int_{e_1}^{e_3}[(x - e_1)(x - e_2)(x - e_3)]^{-1/2}dx.$$ My understanding is that to actually calculate these integrals, one performs a transformation which turns them into complete elliptic integrals of the first kind $$K(k) = \int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1 - k^2\sin^2\theta}},$$ which are given by the Gaussian hypergeometric series. Does anyone have a reference for the change of variables required to convert the period integrals into the elliptic integral? Thanks!

Answer 1

Not sure this is want you want, but

$\sin \theta = t, d \theta = \frac{dt}{ \sqrt{1-t^2}}$ so you get $$\int_0^{\pi / 2} \frac{d \theta}{\sqrt{1-k^2 \sin^2 \theta}} = \int_0^1 \frac{dt}{\sqrt{(1-k^2 t^2)(1-t^2)}}$$ Then $t = \sqrt{u}, dt = \frac{du}{2 \sqrt{u}}$ $$\int_0^1 \frac{dt}{\sqrt{(1-k^2 t^2)(1-t^2)}}=\int_0^1 \frac{du}{2\sqrt{u(1-k^2 u)(1-u)}}$$ With the correct change of variable you should be able to reduce $(x-e_1)(x-e_2)(x-e^3)$ to $u(1-k^2 u)(1-u)$.

Also take a look at "the Pentagonal Number Theorem and Modular Forms" for a larger picture.