EVALUATION OF COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND AT SINGULAR MODULI

Abstract

The complete elliptic integral of the first kind $K(k)$ is defined for $0 \lt k \lt 1$ by \[ K(k) := \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}. \] The real number $k$ is called the modulus of the elliptic integral. The complementary modulus is $k' = (1-k^2)^{\frac{1}{2}}$ ($0 \lt k' \lt 1$). Let $\lambda$ be a positive integer. The equation \[ K(k') = \sqrt{\lambda} K(k) \] defines a unique real number $k(\lambda)$ ($0 \lt k(\lambda) \lt 1$) called the singular modulus of $K(k)$. Let $H(D)$ denote the form class group of discriminant $D$. Let $d$ be the discriminant $-4 \lambda$. Using some recent results of the authors on values of the Dedekind eta function at quadratic irrationalities, a formula is given for the singular modulus $k(\lambda)$ in terms of quantities depending upon $H(4d)$ if $\lambda \equiv 0 \pmod{2}$; $H(d)$ and $H(4d)$ if $\lambda \equiv 1 \pmod{4}$; $H(d/4)$ and $H(4d)$ if $\lambda \equiv 3 \pmod{4}$. Similarly a formula is given for the complete elliptic integral $K[\sqrt{\lambda}] := K(k(\lambda))$ in terms of quantities depending upon $H(d)$ and $H(4d)$ if $\lambda \equiv 0 \pmod{2}$; $H(d)$ if $\lambda \equiv 1 \pmod{4}$; $H(d/4)$ and $H(d)$ if $\lambda \equiv 3 \pmod{4}$. As an example the complete elliptic integral $K[\sqrt{17}]$ is determined explicitly in terms of gamma values.