0028: Part 7, Singular Moduli and Dedekind eta function

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Part 7: Four Singular Moduli and the Complete Elliptic Integral of the First Kind

by Tito Piezas III

Abstract: Formulas for four kinds of singular moduli will be given in terms of the Dedekind eta function η(τ) at arguments τ = 1/p√-m. However, at τ = (1+√-d)/2, these can also be used to compute the complete elliptic integral of the first kind K(k_d).

I. Introduction

II. Four Singular Moduli

III. Complete Elliptic Integral of the First Kind

I. Introduction

In his second letter to Hardy, Ramanujan solved the equation involving the hypergeometric function ₂F₁(a,b;c;z),

using the amazing radical,

δ = (√15-√14)² (8-3√7)² (2-√3)² (6-√35)² (1-√2)⁴ (3-√10)⁴ (4-√15)⁴ (√7-√6)⁴ ≈ 2.7066 x 10^(-19)

Note that these involve powers of fundamental units. For example,

(√15-√14)² = 29-2√210

and {x, y} = {29, 2} is the fundamental solution to the Pell equation x²-210y² = 1. There is a general formula for δ, and similar singular moduli, using the Dedekind eta function to be given here. (However, to express them as products of units as Ramanujan ingeniously did is another matter entirely.)

II. Singular moduli

Solutions to the following four equations,

are given by,

or equivalently,

where j(τ) is the j-function, η(τ) is the Dedekind eta function, and,

for p = {1, 2, 3, 4}, respectively. (The variables {α, β, γ, δ} and the r_p(τ) are also discussed in the previous article, Part 6: General forms for Ramanujan’s pi formulas.) For example, the solution to the equations,

are given, respectively, by,

p = 1, τ = (1/1)√-7 and r₁(τ) = 255³.

p = 2, τ = (1/2)√-58 and r₂(τ) = 396⁴.

p = 3, τ = (1/3)√-15 and r₃(τ) = 15³.

p = 4, τ = (1/4)√-28 and r₄(τ) = 2¹².

such that the moduli are,

Ramanujan’s example is then p = 4, τ = (1/4)√-840, and its r₄(τ) is easily calculated using the Dedekind eta function, though to express it as radicals ab initio is not so easy. In general, for m a positive integer, then α, β, γ, δ, are algebraic numbers.

III. Complete Elliptic Integral of the First Kind

The complete elliptic integral of the first kind K(k) is defined for 0 < k < 1 by,

or equivalently,

But it can also be expressed in terms of α, β, δ (missing the third moduli, γ) as,

where τ is now restricted to the form,

for positive d above a bound, otherwise r_p(τ) may be zero. For example, for p = 1, then at least d > 3, since for d = 3, it is well-known that r₁(τ) = j(τ) = 0. But, say, for d = 163, then,

where,

and x is the real root of the cubic, x³-6x²+4x-2 = 0. Similarly, for p = 2, and d = 37,

where,