The periods of the Weierstrass function $\wp(z)$

Question

Is it true that the periods $\omega_1$, $\omega_2$ of $\wp(z)$ are $\omega_1 = 4K$ and $\omega_2 = 4iK'$, respectively? Here, $K = K(k)$ is the complete elliptic integral of the first kind, and $K' = K(k')$ with $k' = \sqrt{1 - k^2}$, where $0 < k < 1$ is the elliptic modulus. I could not find these stated explicitly anywhere, but I deduced that $\omega_1 = 4K$ and $\omega_2 = 4iK'$ from some relations in Greenhill's book on elliptic functions. It seems to be correct, but it bothers me that I could not find a single book on elliptic functions (not even Whittaker and Watson) that states that $\omega_1 = 4K$ and $\omega_2 = 4iK'$. Any help would be appreciated.

Answer 1

It is not true unless I misunderstand your question. The periods $\omega_1,\omega_2$ form a pair of generic complex numbers, which cannot be parameterized with a single elliptic modulus $k$. Note in particular that simultaneous rescaling $(\omega_1,\omega_2)\mapsto (\lambda\omega_1,\lambda\omega_2)$ does not change $k$.

Geometrically, $\wp(z)$ describes uniformization of elliptic curve realized as a two-sheeted covering of $\mathbb P^1$. The modulus $k$ is related to the anharmonic ratio of the branch points $e_1,e_2,e_3,\infty$, which remains invariant under rescalings.

For explicit relation between $e$'s and $\omega$'s, see here.