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 < k < 1 by K(k): = ∫0π/2 dθ/√1-k 2 sin2 θ. The real number k is called the modulus of the elliptic integral. The complementary modulus is k′ = (1 - k 2)1/2 (0 < k′ < 1). Let λ be a positive integer. The equation K(k′) = √λK(k) defines a unique real number k(λ) (0 < k(λ) < 1) called the singular modulus of K(k), Let H(D) denote the form class group of discriminant D. Let d be the discriminant -4λ. 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(λ) in terms of quantities depending upon H(4d) if λ = 0 (mod 2); H(d) and H(4d) if λ = 1 (mod 4); H(d/4) and H(4d) if λ = 3 (mod 4). Similarly a formula is given for the complete elliptic integral K[√λ] := K(k(λ)) in terms of quantities depending upon H(d) and H(4d) if λ = 0 (mod 2); H(d) if λ = 1 (mod 4); H(d/4) and H(d) if λ = 3 (mod 4). As an example the complete elliptic integral K[√17] is determined explicitly in terms of gamma values.