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Legendre’s Singular Modulus Mark B. Villarino Escuela de Matem´atica, Universidad de Costa Rica, 11501 San Jos´e, Costa Rica 1 Introduction Two characteristics of mathematics charm and delight most professional mathematicians. The first is its historical continuity. For example, although Euclid created his proofs of Pythagoras’s theorem [9, I.47] and of the infinitude of primes [9, IX.20] some 2300 years ago, they remain as fresh, as compelling, and as beautiful today as they were when he first wrote them down. The second is the way that seemingly disparate areas of mathematics reveal deep and unsuspected relationships. For example the number π appears in a myriad distant parts of mathematics. One might say that π is ubiquitous. A personal favorite of ours is the fact, discovered by Dirichlet [12, Thm. 332], that the probability that two integers taken at random are relatively prime 6 is . What on earth does π, the universal ratio of the circumference to π2 the diameter of any circle, have to do with the common divisors of two integers taken at random? Dirichlet’s result shows that the relationship is profound. Indeed it is based on Euler’s solution to the Basel problem, i.e., arXiv:2002.07250v2 [math.HO] 3 Apr 2020 1 π2 ∞ that n=1 n2 = 6 , which, in turn, is based on the fact that the non zero roots of the transcendental equation sin x = 0 are π, 2π, 3π,... OurP present paper is devoted the unexpected± and± fascinating± ubiquity π of Legendre’s relatively unknown (third) singular modulus k := sin 12 = 1 2 √3 (see the definition below). These instances include: 2 − p Legendre’s original proof of the first appearance in the history of math- • ematics of a singular modulus; 1 Ramanujan’s formula for the arc length of an ellipse of eccentricity • π sin 12 ; the three-body choreography on a lemniscate. • We also briefly mention random walks on a cubic lattice and simple pen- dulum renormalization. These occurrences are familiar to most workers in these areas but not to the general mathematical public at large. They deserve to be better known since they display the two characteristics we listed above and they exhibit quite beautiful mathematics. We hope that our paper makes them easily available. 2 Legendre’s Singular Modulus In 1811, the famous french mathematician Adrien-Marie Legendre published the following notable result [14, p. 60], which we present in a form close to the spirit of Ramanujan. Theorem 2.1. If 1 2 1 3 2 1 3 5 2 f(α) := 1 + α + · α2 + · · α3 + 2 2 4 2 4 6 ··· · · · and f(1 α)= √3 f(α), − · then π 1 α = sin2 = (2 √3). 12 4 − We we note that the series converges for 1 6 α< 1 and that the quotient f(1 α) − − decreases monotonically from to 1 as α increases from 0 to 1. Thus f(α) ∞ there is a unique α such that f(1 α)= √3 f(α) is true. − · We now formulate Legendre’s own statement [14, p. 59]. Let 0 6 k 6 1 and π/2 dθ 2 K(k) := and K′ := K(√1 k ). 2 0 1 k2 sin θ − Z − Then, if we expand thep integrand by the binomial theorem, integrate term by term, and write k2 for α, we obtain Legendre’s original form. 2 Theorem 2.2. K (k) π 1 ′ = √3 = k2 = sin2 = (2 √3) (2.1) K(k) ⇒ 12 4 − The equation (2.1) is the first example of a singular modulus and we define the concept below. Definition 2.1. If N is a positive integer and the following equation holds: K (k) ′ = √N, (2.2) K(k) then k is called the singular modulus for N. There is an enormous literature dealing with singular moduli. Today the branch of mathematics which studies them is called complex multiplication [8]. Legendre’s relation (2.1) is the first published explicit example of it. The great German mathematician, David Hilbert, remarked that the the- ory of complex multiplication ”...which brings together number theory and analysis, is not only the most beautiful part of mathematics, but of all of science.” [18], (p. 200) The theory was born when Abel [1] published the following general theorem. Theorem 2.3. Whenever K a + b√N ′ = K c + d√N where a,b,c,d are integers, k is the root of an algebraic equation with integral coefficients which is solvable by radicals. The group of the equation turns out to be abelian, which is why it is solvable by radicals. The splitting field is therefore called an abelian ex- tension. Thus the algebraic nature of the modulus is deeply related to its arithmetic properties. Indeed, Gauss’ theory of cyclotomy, in which the cy- clotomic equation has an abelian group, and thus is solvable by radicals, lead Abel by analogy to a corresponding theory of using a ruler and compass to divide a lemniscate into equal arcs, and thus the above theorem. Kronecker suggested that since the roots of unity, which are values of the exponential function, generate abelian extensions of the rational number field, perhaps 3 also elliptic functions could generate abelian extensions of quadratic num- ber fields, his so-called ”Jugentraum.” Years of work showed that Kronecker was basically right and this theory of ”singular moduli,” a term introduced by Kronecker, would create wonderful mathematics. A simple example is the Heegner/Stark proof that there are only nine imaginary quadratic num- ber fields which admit unique factorization, a problem posed by Gauss some 170 years earlier. Or the transcendental proof of the biquadratic reciprocity theorem. Finally it was natural to consider abelian extensions of arbitrary algebraic number fields which lead to today’s profound theories called ”class field theory.” It is amazing that transcendental functions lead to the solu- tion of deep problems in discrete number theory and commutative algebra, all brought about by Legendre’s original singular modulus and Abel’s brilliant generalization. All published proofs of (2.1), except for Legendre’s own, either use Ja- cobi’s theory of the cubic transformation and the modular equation of degree three [5, p. 188], or the complex variable proof given in [22, pp. 525–526] or [4, p. 92]. Legendre’s original proof, on the other hand, is completely elementary and we have not been able to find a presentation of it in the literature. Perhaps the reason is that some fifteen years later Jacobi and Abel made Legendre’s almost forty years of work obsolete by studying the elliptic functions instead of integrals and much of his work was subsequently neglected. Still, it is a brilliant tour de force in first-year integral calculus and deserves to be better known. Unfortunately it is seemingly unmotivated. Indeed, we suggest that Legendre discovered his result by accident and then developed his proof, which is a verification. Even so, we will present it here. It is based on his theory of the bisection and trisection of elliptic integrals. To this end we will first briefly review those elementary properties of (real) elliptic integrals which Legendre uses in his proof. Then will give a detailed presentation of Legendre’s proof itself. 2.1 Elliptic Integrals Definition 2.2. Let k be a real number such that 0 6 k 6 1 and Φ a real 6 6 π number such that 0 Φ 2 . Then Φ dφ F (Φ,k) := 2 0 1 k2 sin φ Z − p 4 is called an elliptic integral (of the first kind), Φ is the amplitude and k is the modulus. Usually we write F (Φ) for brevity if the modulus is not important. Legendre proposed the following trisection problem: If k is given, it is required to find the amplitude Φ which solves the following equation: Φ dφ 1 π/2 dψ K F (Φ) = . (2.3) 2 2 ≡ 0 1 k2 sin φ 3 0 1 k2 sin ψ ≡ 3 Z − Z − Legendre solvesp it by proving (after Euler)p that F (Φ) satisfies an addition theorem [14, p. 20]: namely F (Φ) + F (Ψ) = F (µ), (2.4) where cos µ = cosΦcosΨ sin Φ sin Ψ 1 k2 sin2 µ. − − π q Taking µ = 2 and F (Ψ) := F (Φ)+ F (Φ) in (2.4), after some algebra and trigonometric reduction Legendre [14, p. 29] finds the following result. Theorem 2.4. Let k be fixed. Then x := sin Φ in (2.3) is a root of k2x4 2k2x3 +2x 1=0. (2.5) − − He then investigates two special cases. 1 √ π Corollary 2.5. (a) If k := 2 2+ 3 = cos 12 , then p 2 tanΦ = s√3 and vice versa. (b) If k := 1 2 √3 = sin π , then 2 − 12 p 2+ √3 cosΦ = (22/3 1) − s √3 and vice versa. 5 The proofs are elementary and only involve high-school algebra and trigonom- etry. Along the way, Legendre also solves the general bisection problem [14, p. 25]. (We follow Beenakker [2].) Theorem 2.6. If sin θ sin Φ = 2 , ∆(θ) := 1 k2 sin2 θ, 1 + 1 ∆(θ) − 2 2 p q then 1 F (Φ,k)= 2 F (θ,k). It is convenient to prove the following technical lemma (we follow Bowman [4, p. 91]). Lemma 2.7. If T := 1 + 2t2 cos2α + t4, then x 2x/(1+x2) dt ∞ dt 1 dy = = (2.6) 2 2 2 0 √T 1/x √T 2 0 (1 y )(1 k y ) Z Z Z − − π where 0 <x< 1 and 0 <α< 2 and k := sin αp.

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