Modular lambda function

In mathematics, the modular lambda function λ(τ) is a highly symmetric Holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group &Gamma;(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve $$\mathbb{C}/\langle 1, \tau \rangle$$, where the map is defined as the quotient by the [&minus;1] involution.

The q-expansion, where $$q = e^{\pi i \tau}$$ is the nome, is given by:


 * $$ \lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots$$.

By symmetrizing the lambda function under the canonical action of the symmetric group S3 on X(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group $$\operatorname{SL}_2(\mathbb{Z})$$, and it is in fact Klein's modular j-invariant.

Modular properties
The function $$ \lambda(\tau) $$ is invariant under the group generated by


 * $$ \tau \mapsto \tau+2 \ ;\ \tau \mapsto \frac{\tau}{1-2\tau} \ . $$

The generators of the modular group act by


 * $$ \tau \mapsto \tau+1 \ :\ \lambda \mapsto \frac{\lambda}{\lambda-1} \, ;$$
 * $$ \tau \mapsto -\frac{1}{\tau} \ :\ \lambda \mapsto 1 - \lambda \ . $$

Consequently, the action of the modular group on $$ \lambda(\tau) $$ is that of the anharmonic group, giving the six values of the cross-ratio:


 * $$ \left\lbrace { \lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{1}{\lambda}, \frac{\lambda}{\lambda-1}, 1-\lambda } \right\rbrace \ .$$

Relations to other functions
It is the square of the elliptic modulus, that is, $$\lambda(\tau)=k^2(\tau)$$. In terms of the Dedekind eta function $$\eta(\tau)$$ and theta functions,


 * $$ \lambda(\tau) = \Bigg(\frac{\sqrt{2}\,\eta(\tfrac{\tau}{2})\eta^2(2\tau)}{\eta^3(\tau)}\Bigg)^8 = \frac{16}{\left(\frac{\eta(\tau/2)}{\eta(2\tau)}\right)^8 + 16} =\frac{\theta_2^4(\tau)}{\theta_3^4(\tau)} $$

and,


 * $$ \frac{1}{\big(\lambda(\tau)\big)^{1/4}}-\big(\lambda(\tau)\big)^{1/4} = \frac{1}{2}\left(\frac{\eta(\tfrac{\tau}{4})}{\eta(\tau)}\right)^4 = 2\,\frac{\theta_4^2(\tfrac{\tau}{2})}{\theta_2^2(\tfrac{\tau}{2})}$$

where


 * $$\theta_2(\tau) =\sum_{n=-\infty}^\infty e^{\pi i\tau (n+1/2)^2}$$


 * $$\theta_3(\tau) = \sum_{n=-\infty}^\infty e^{\pi i\tau n^2}$$


 * $$\theta_4(\tau) = \sum_{n=-\infty}^\infty (-1)^n e^{\pi i\tau n^2} $$

In terms of the half-periods of Weierstrass's elliptic functions, let $$[\omega_1,\omega_2]$$ be a fundamental pair of periods with $$\tau=\frac{\omega_2}{\omega_1}$$.


 * $$ e_1 = \wp\left(\frac{\omega_1}{2}\right), \quad e_2 = \wp\left(\frac{\omega_2}{2}\right),\quad e_3 = \wp\left(\frac{\omega_1+\omega_2}{2}\right) $$

we have


 * $$ \lambda = \frac{e_3-e_2}{e_1-e_2} \, . $$

Since the three half-period values are distinct, this shows that $$\lambda$$ does not take the value 0 or 1.

The relation to the j-invariant is


 * $$ j(\tau) = \frac{256(1-\lambda(1-\lambda))^3}{(\lambda(1-\lambda))^2} = \frac{256(1-\lambda+\lambda^2)^3}{\lambda^2 (1-\lambda)^2} \ . $$

which is the j-invariant of the elliptic curve of Legendre form $$y^2=x(x-1)(x-\lambda)$$

Given $$m\in\mathbb{C}\setminus\{0,1\}$$, let
 * $$\tau=i\frac{K\{1-m\}}{K\{m\}}$$

where $$K$$ is the complete elliptic integral of the first kind with parameter $$m=k^2$$. Then
 * $$\lambda (\tau)=m.$$

Modular equations
The modular equation of degree $$p$$ (where $$p$$ is a prime number) is an algebraic equation in $$\lambda (p\tau)$$ and $$\lambda (\tau)$$. If $$\lambda (p\tau)=u^8$$ and $$\lambda (\tau)=v^8$$, the modular equations of degrees $$p=2,3,5,7$$ are, respectively,
 * $$(1+u^4)^2v^8-4u^4=0,$$
 * $$u^4-v^4+2uv(1-u^2v^2)=0,$$
 * $$u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0,$$
 * $$(1-u^8)(1-v^8)-(1-uv)^8=0.$$

The quantity $$v$$ (and hence $$u$$) can be thought of as a holomorphic function on the upper half-plane $$\operatorname{Im}\tau>0$$:
 * $$\begin{align}v&=\prod_{k=1}^\infty \tanh\frac{(k-1/2)\pi i}{\tau}=\sqrt{2}e^{\pi i\tau/8}\frac{\sum_{k\in\mathbb{Z}}e^{(2k^2+k)\pi i\tau}}{\sum_{k\in\mathbb{Z}}e^{k^2\pi i\tau}}\\

&=\cfrac{\sqrt{2}e^{\pi i\tau/8}}{1+\cfrac{e^{\pi i\tau}}{1+e^{\pi i\tau}+\cfrac{e^{2\pi i\tau}}{1+e^{2\pi i\tau}+\cfrac{e^{3\pi i\tau}}{1+e^{3\pi i\tau}+\ddots}}}}\end{align}$$ Since $$\lambda(i)=1/2$$, the modular equations can be used to give algebraic values of $$\lambda(pi)$$ for any prime $$p$$. The algebraic values of $$\lambda(ni)$$ are also given by
 * $$\lambda (ni)=\prod_{k=1}^{n/2} \operatorname{sl}^8\frac{(2k-1)\varpi}{2n}\quad (n\,\text{even})$$
 * $$\lambda (ni)=\frac{1}{2^n}\prod_{k=1}^{n-1} \left(1-\operatorname{sl}^2\frac{k\varpi}{n}\right)^2\quad (n\,\text{odd})$$

where $$\operatorname{sl}$$ is the lemniscate sine and $$\varpi$$ is the lemniscate constant.

Definition and computation of lambda-star
The function $$\lambda^*(x)$$ (where $$x\in\mathbb{R}^+$$) gives the value of the elliptic modulus $$k$$, for which the complete elliptic integral of the first kind $$K(k)$$ and its complementary counterpart $$K(\sqrt{1-k^2})$$ are related by following expression:


 * $$\frac{K\left[\sqrt{1-\lambda^*(x)^2}\right]}{K[\lambda^*(x)]} = \sqrt{x}$$

The values of $$\lambda^*(x)$$ can be computed as follows:


 * $$\lambda^*(x) = \frac{\theta^2_2(i\sqrt{x})}{\theta^2_3(i\sqrt{x})} $$


 * $$\lambda^*(x) = \left[\sum_{a=-\infty}^\infty\exp[-(a+1/2)^2\pi\sqrt{x}]\right]^2\left[\sum_{a=-\infty}^\infty\exp(-a^2\pi\sqrt{x})\right]^{-2} $$


 * $$\lambda^*(x) = \left[\sum_{a=-\infty}^\infty\operatorname{sech}[(a+1/2)\pi\sqrt{x}]\right]\left[\sum_{a=-\infty}^\infty\operatorname{sech}(a\pi\sqrt{x})\right]^{-1} $$

The functions $$\lambda^*$$ and $$\lambda$$ are related to each other in this way:


 * $$\lambda^*(x) = \sqrt{\lambda(i\sqrt{x})}$$

Properties of lambda-star
Every $$\lambda^*$$ value of a positive rational number is a positive algebraic number:


 * $$\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.$$

$$K(\lambda^*(x))$$ and $$E(\lambda^*(x))$$ (the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any $$x\in\mathbb{Q}^+$$, as Selberg and Chowla proved in 1949.

The following expression is valid for all $$n \in \mathbb{N}$$:


 * $$\sqrt{n} = \sum_{a = 1}^{n} \operatorname{dn}\left[\frac{2a}{n}K\left[\lambda^*\left(\frac{1}{n}\right)\right];\lambda^*\left(\frac{1}{n}\right)\right] $$

where $$\operatorname{dn}$$ is the Jacobi elliptic function delta amplitudinis with modulus $$k$$.

By knowing one $$\lambda^*$$ value, this formula can be used to compute related $$\lambda^*$$ values:


 * $$\lambda^*(n^2x) = \lambda^*(x)^n\prod_{a=1}^{n}\operatorname{sn}\left\{\frac{2a-1}{n}K[\lambda^*(x)];\lambda^*(x)\right\}^2 $$

where $$n\in\mathbb{N}$$ and $$\operatorname{sn}$$ is the Jacobi elliptic function sinus amplitudinis with modulus $$k$$.

Further relations:


 * $$\lambda^*(x)^2 + \lambda^*(1/x)^2 = 1 $$


 * $$[\lambda^*(x)+1][\lambda^*(4/x)+1] = 2 $$


 * $$\lambda^*(4x) = \frac{1-\sqrt{1-\lambda^*(x)^2}}{1+\sqrt{1-\lambda^*(x)^2}} = \tan\left\{\frac{1}{2}\arcsin[\lambda^*(x)]\right\}^2 $$


 * $$\lambda^*(x) - \lambda^*(9x) = 2[\lambda^*(x)\lambda^*(9x)]^{1/4} - 2[\lambda^*(x)\lambda^*(9x)]^{3/4}$$

$$\begin{align} & a^{6}-f^{6} = 2af +2a^5f^5\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(f = \left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{1/12}\right) \\ &a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(b = \left[\frac{2\lambda^*(49x)}{1-\lambda^*(49x)^2}\right]^{1/12}\right) \\

& a^{12}-c^{12} = 2\sqrt{2}(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2+2a^4c^4)\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(c = \left[\frac{2\lambda^*(121x)}{1-\lambda^*(121x)^2}\right]^{1/12}\right) \\

& (a^2-d^2)(a^4+d^4-7a^2d^2)[(a^2-d^2)^4-a^2d^2(a^2+d^2)^2] = 8ad+8a^{13}d^{13}\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) &\left(d = \left[\frac{2\lambda^*(169x)}{1-\lambda^*(169x)^2}\right]^{1/12}\right) \end{align} $$

Lambda-star values of integer numbers of 4n-3-type:


 * $$\lambda^*(1) = \frac{1}{\sqrt{2}}$$


 * $$\lambda^*(5) = \sin\left[\frac{1}{2}\arcsin\left(\sqrt{5}-2\right)\right]$$


 * $$\lambda^*(9) = \frac{1}{2}(\sqrt{3}-1)(\sqrt{2}-\sqrt[4]{3})$$


 * $$\lambda^*(13) = \sin\left[\frac{1}{2}\arcsin(5\sqrt{13}-18)\right]$$


 * $$\lambda^*(17) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{64}\left(5+\sqrt{17}-\sqrt{10\sqrt{17}+26}\right)^3\right]\right\}$$


 * $$\lambda^*(21) = \sin\left\{\frac{1}{2}\arcsin[(8-3\sqrt{7})(2\sqrt{7}-3\sqrt{3})]\right\}$$


 * $$\lambda^*(25) = \frac{1}{\sqrt{2}}(\sqrt{5}-2)(3-2\sqrt[4]{5})$$


 * $$\lambda^*(33) = \sin\left\{\frac{1}{2}\arcsin[(10-3\sqrt{11})(2-\sqrt{3})^3]\right\}$$


 * $$\lambda^*(37) = \sin\left\{\frac{1}{2}\arcsin[(\sqrt{37}-6)^3]\right\}$$


 * $$\lambda^*(45) = \sin\left\{\frac{1}{2}\arcsin[(4-\sqrt{15})^2(\sqrt{5}-2)^3]\right\}$$


 * $$\lambda^*(49) = \frac{1}{4}(8+3\sqrt{7})(5-\sqrt{7}-\sqrt[4]{28})\left(\sqrt{14}-\sqrt{2}-\sqrt[8]{28}\sqrt{5-\sqrt{7}}\right)$$


 * $$\lambda^*(57) = \sin\left\{\frac{1}{2}\arcsin[(170-39\sqrt{19})(2-\sqrt{3})^3]\right\}$$


 * $$\lambda^*(73) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{64}\left(45+5\sqrt{73}-3\sqrt{50\sqrt{73}+426}\right)^3\right]\right\}$$

Lambda-star values of integer numbers of 4n-2-type:


 * $$\lambda^*(2) = \sqrt{2}-1$$


 * $$\lambda^*(6) = (2-\sqrt{3})(\sqrt{3}-\sqrt{2})$$


 * $$\lambda^*(10) = (\sqrt{10}-3)(\sqrt{2}-1)^2$$


 * $$\lambda^*(14) = \tan\left\{\frac{1}{2}\arctan\left[\frac{1}{8}\left(2\sqrt{2}+1-\sqrt{4\sqrt{2}+5}\right)^3\right]\right\}$$


 * $$\lambda^*(18) = (\sqrt{2}-1)^3(2-\sqrt{3})^2$$


 * $$\lambda^*(22) = (10-3\sqrt{11})(3\sqrt{11}-7\sqrt{2})$$


 * $$\lambda^*(30) = \tan\left\{\frac{1}{2}\arctan[(\sqrt{10}-3)^2(\sqrt{5}-2)^2]\right\}$$


 * $$\lambda^*(34) = \tan\left\{\frac{1}{4}\arcsin\left[\frac{1}{9}(\sqrt{17}-4)^2\right]\right\}$$


 * $$\lambda^*(42) = \tan\left\{\frac{1}{2}\arctan[(2\sqrt{7}-3\sqrt{3})^2(2\sqrt{2}-\sqrt{7})^2]\right\}$$


 * $$\lambda^*(46) = \tan\left\{\frac{1}{2}\arctan\left[\frac{1}{64}\left(3+\sqrt{2}-\sqrt{6\sqrt{2}+7}\right)^6\right]\right\}$$


 * $$\lambda^*(58) = (13\sqrt{58}-99)(\sqrt{2}-1)^6$$


 * $$\lambda^*(70) = \tan\left\{\frac{1}{2}\arctan[(\sqrt{5}-2)^4(\sqrt{2}-1)^6]\right\}$$


 * $$\lambda^*(78) = \tan\left\{\frac{1}{2}\arctan[(5\sqrt{13}-18)^2(\sqrt{26}-5)^2]\right\}$$


 * $$\lambda^*(82) = \tan\left\{\frac{1}{4}\arcsin\left[\frac{1}{4761}(8\sqrt{41}-51)^2\right]\right\}$$

Lambda-star values of integer numbers of 4n-1-type:


 * $$\lambda^*(3) = \frac{1}{2\sqrt{2}}(\sqrt{3}-1)$$


 * $$\lambda^*(7) = \frac{1}{4\sqrt{2}}(3-\sqrt{7})$$


 * $$\lambda^*(11) = \frac{1}{8\sqrt{2}}(\sqrt{11}+3)\left(\frac{1}{3}\sqrt[3]{6\sqrt{3}+2\sqrt{11}}-\frac{1}{3}\sqrt[3]{6\sqrt{3}-2\sqrt{11}}+\frac{1}{3}\sqrt{11}-1\right)^4$$


 * $$\lambda^*(15) = \frac{1}{8\sqrt{2}}(3-\sqrt{5})(\sqrt{5}-\sqrt{3})(2-\sqrt{3})$$


 * $$\lambda^*(19) = \frac{1}{8\sqrt{2}}(3\sqrt{19}+13)\left[\frac{1}{6}(\sqrt{19}-2+\sqrt{3})\sqrt[3]{3\sqrt{3}-\sqrt{19}}-\frac{1}{6}(\sqrt{19}-2-\sqrt{3})\sqrt[3]{3\sqrt{3}+\sqrt{19}}-\frac{1}{3}(5-\sqrt{19})\right]^4$$


 * $$\lambda^*(23) = \frac{1}{16\sqrt{2}}(5+\sqrt{23})\left[\frac{1}{6}(\sqrt{3}+1)\sqrt[3]{100-12\sqrt{69}}-\frac{1}{6}(\sqrt{3}-1)\sqrt[3]{100+12\sqrt{69}}+\frac{2}{3}\right]^4$$


 * $$\lambda^*(27) = \frac{1}{16\sqrt{2}}(\sqrt{3}-1)^3\left[\frac{1}{3}\sqrt{3}(\sqrt[3]{4}-\sqrt[3]{2}+1)-\sqrt[3]{2}+1\right]^4$$


 * $$\lambda^*(39) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{16}\left(6-\sqrt{13}-3\sqrt{6\sqrt{13}-21}\right)\right]\right\}$$


 * $$\lambda^*(55) = \sin\left\{\frac{1}{2}\arcsin\left[\frac{1}{512}\left(3\sqrt{5}-3-\sqrt{6\sqrt{5}-2}\right)^3\right]\right\}$$

Lambda-star values of integer numbers of 4n-type:


 * $$\lambda^*(4) = (\sqrt{2}-1)^2$$


 * $$\lambda^*(8) = \left(\sqrt{2}+1-\sqrt{2\sqrt{2}+2}\right)^2$$


 * $$\lambda^*(12) = (\sqrt{3}-\sqrt{2})^2(\sqrt{2}-1)^2$$


 * $$\lambda^*(16) = (\sqrt{2}+1)^2(\sqrt[4]{2}-1)^4$$


 * $$\lambda^*(20) = \tan\left[\frac{1}{4}\arcsin(\sqrt{5}-2)\right]^2$$


 * $$\lambda^*(24) = \tan\left\{\frac{1}{2}\arcsin[(2-\sqrt{3})(\sqrt{3}-\sqrt{2})]\right\}^2$$


 * $$\lambda^*(28) = (2\sqrt{2}-\sqrt{7})^2(\sqrt{2}-1)^4$$


 * $$\lambda^*(32) = \tan\left\{\frac{1}{2}\arcsin\left[\left(\sqrt{2}+1-\sqrt{2\sqrt{2}+2}\right)^2\right]\right\}^2$$

Lambda-star values of rational fractions:


 * $$\lambda^*\left(\frac{1}{2}\right) = \sqrt{2\sqrt{2}-2}$$


 * $$\lambda^*\left(\frac{1}{3}\right) = \frac{1}{2\sqrt{2}}(\sqrt{3}+1)$$


 * $$\lambda^*\left(\frac{2}{3}\right) = (2-\sqrt{3})(\sqrt{3}+\sqrt{2})$$


 * $$\lambda^*\left(\frac{1}{4}\right) = 2\sqrt[4]{2}(\sqrt{2}-1)$$


 * $$\lambda^*\left(\frac{3}{4}\right) = \sqrt[4]{8}(\sqrt{3}-\sqrt{2})(\sqrt{2}+1)\sqrt{(\sqrt{3}-1)^3}$$


 * $$\lambda^*\left(\frac{1}{5}\right) = \frac{1}{2\sqrt{2}}\left(\sqrt{2\sqrt{5}-2}+\sqrt{5}-1\right)$$


 * $$\lambda^*\left(\frac{2}{5}\right) = (\sqrt{10}-3)(\sqrt{2}+1)^2$$


 * $$\lambda^*\left(\frac{3}{5}\right) = \frac{1}{8\sqrt{2}}(3+\sqrt{5})(\sqrt{5}-\sqrt{3})(2+\sqrt{3})$$


 * $$\lambda^*\left(\frac{4}{5}\right) = \tan\left[\frac{\pi}{4}-\frac{1}{4}\arcsin(\sqrt{5}-2)\right]^2$$

Ramanujan's class invariants
Ramanujan's class invariants $$G_n$$ and $$g_n$$ are defined as
 * $$G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right),$$
 * $$g_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1-e^{-(2k+1)\pi\sqrt{n}}\right),$$

where $$n\in\mathbb{Q}^+$$. For such $$n$$, the class invariants are algebraic numbers. For example


 * $$g_{58}=\sqrt{\frac{5+\sqrt{29}}{2}}, \quad g_{190}=\sqrt{(\sqrt{5}+2)(\sqrt{10}+3)}.$$

Identities with the class invariants include


 * $$G_n=G_{1/n},\quad g_{n}=\frac{1}{g_{4/n}},\quad g_{4n}=2^{1/4}g_nG_n.$$

The class invariants are very closely related to the Weber modular functions $$\mathfrak{f}$$ and $$\mathfrak{f}_1$$. These are the relations between lambda-star and the class invariants:


 * $$G_n = \sin\{2\arcsin[\lambda^*(n)]\}^{-1/12} = 1\Big /\left[\sqrt[12]{2\lambda^*(n)}\sqrt[24]{1-\lambda^*(n)^2}\right] $$


 * $$g_n = \tan\{2\arctan[\lambda^*(n)]\}^{-1/12} = \sqrt[12]{[1-\lambda^*(n)^2]/[2\lambda^*(n)]} $$


 * $$\lambda^*(n) = \tan\left\{ \frac{1}{2}\arctan[g_n^{-12}]\right\} = \sqrt{g_n^{24}+1}-g_n^{12} $$

Little Picard theorem
The lambda function is used in the original proof of the Little Picard theorem, that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879. Suppose if possible that f is entire and does not take the values 0 and 1. Since λ is holomorphic, it has a local holomorphic inverse ω defined away from 0,1,∞. Consider the function z → ω(f(z)). By the Monodromy theorem this is holomorphic and maps the complex plane C to the upper half plane. From this it is easy to construct a holomorphic function from C to the unit disc, which by Liouville's theorem must be constant.

Moonshine
The function $$\tau\mapsto 16/\lambda(2\tau) - 8$$ is the normalized Hauptmodul for the group $$\Gamma_0(4)$$, and its q-expansion $$q^{-1} + 20q - 62q^3 + \dots$$,  where $$q=e^{2\pi i\tau }$$, is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

Other



 * Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.


 * Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.


 * Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.