Modular lambda function

File:Modular lambda function in range -3 to 3.png

Modular lambda function in the complex plane.

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 Γ(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 $ℂ/⟨1,\tau ⟩$, where the map is defined as the quotient by the [−1] involution.

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

$\lambda (\tau )=16q-128q^2+704q^3-3072q^4+11488q^5-38400q^6+\dots$. (sequence A115977 in the OEIS)

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 ${\mathrm{SL}}_2(\mathbb{Z})$, and it is in fact Klein's modular j-invariant.

A plot of x→ λ(ix)

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:

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

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,

$\lambda (\tau )=(\frac{\sqrt{2}\eta (\frac{\tau }{2}){\eta }^2(2\tau )}{{\eta }^3(\tau )}{)}^8=\frac{16}{{\left(\frac{\eta (\tau /2)}{\eta (2\tau )}\right)}^8+16}=\frac{{\theta }_2^4(\tau )}{{\theta }_3^4(\tau )}$

and,

$\frac{1}{(\lambda (\tau ){)}^{1/4}}-(\lambda (\tau ){)}^{1/4}=\frac{1}{2}{\left(\frac{\eta (\frac{\tau }{4})}{\eta (\tau )}\right)}^4=2\frac{{\theta }_4^2(\frac{\tau }{2})}{{\theta }_2^2(\frac{\tau }{2})}$

where

${\theta }_2(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\pi i\tau (n+1/2{)}^2}$

${\theta }_3(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\pi i\tau n^2}$

${\theta }_4(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{e}^{\pi i\tau n^2}$

In terms of the half-periods of Weierstrass's elliptic functions, let $[{\omega }_1,{\omega }_2]$ be a fundamental pair of periods with $\tau =\frac{{\omega }_2}{{\omega }_1}$.

$e_1=\mathrm{\wp }\left(\frac{{\omega }_1}{2}\right),e_2=\mathrm{\wp }\left(\frac{{\omega }_2}{2}\right),e_3=\mathrm{\wp }\left(\frac{{\omega }_1+{\omega }_2}{2}\right)$

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 $m\in ℂ\setminus \{0,1\}$, 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,

$(1+u^4{)}^2{v}^8-4u^4=0,$

$u^4-v^4+2uv(1-u^2{v}^2)=0,$

$u^6-v^6+5u^2{v}^2(u^2-v^2)+4uv(1-u^4{v}^4)=0,$

$(1-u^8)(1-v^8)-(1-uv{)}^8=0.$

The quantity $v$ (and hence $u$) can be thought of as a holomorphic function on the upper half-plane $\mathrm{Im}\tau >0$:

$\begin{array}{cc}v& =\prod _{k=1}^{\mathrm{\infty }}\tanh \frac{(k-1/2)\pi i}{\tau }=\sqrt{2}{e}^{\pi i\tau /8}\frac{\sum _{k\in \mathbb{Z}}{e}^{(2k^2+k)\pi i\tau }}{\sum _{k\in \mathbb{Z}}{e}^{k^2\pi i\tau }}\\ & =\frac{\sqrt{2}{e}^{\pi i\tau /8}}{1+\frac{e^{\pi i\tau }}{1+e^{\pi i\tau }+\frac{e^{2\pi i\tau }}{1+e^{2\pi i\tau }+\frac{e^{3\pi i\tau }}{1+e^{3\pi i\tau }+\ddots }}}}\end{array}$

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

$\lambda (ni)=\prod _{k=1}^{n/2}{\mathrm{sl}}^8\frac{(2k-1)\varpi }{2n}(n\text{even})$

$\lambda (ni)=\frac{1}{2^n}\prod _{k=1}^{n-1}{(1-{\mathrm{sl}}^2\frac{k\varpi }{n})}^2(n\text{odd})$

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

Lambda-star

Definition and computation of lambda-star

The function ${\lambda }^{*}(x)$ (where $x\in {ℝ}^{+}$) 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:

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

${\lambda }^{*}(x)={[\sum _{a=-\mathrm{\infty }}^{\mathrm{\infty }}\exp [-(a+1/2{)}^2\pi \sqrt{x}]]}^2{[\sum _{a=-\mathrm{\infty }}^{\mathrm{\infty }}\exp (-a^2\pi \sqrt{x})]}^{-2}$

${\lambda }^{*}(x)=[\sum _{a=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{sech}[(a+1/2)\pi \sqrt{x}]]{[\sum _{a=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{sech}(a\pi \sqrt{x})]}^{-1}$

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:

${\lambda }^{*}(x)\in {\bar{\mathbb{Q}}}_{+}\mathrm{\forall }x\in {\mathbb{Q}}^{+}.$

$K({\lambda }^{*}(x))$ 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 $n\in \mathbb{N}$:

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

where $\mathrm{dn}$ 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:

${\lambda }^{*}(n^{2}x)={\lambda }^{*}(x{)}^n\prod _{a=1}^n\mathrm{sn}{\{\frac{2a-1}{n}K[{\lambda }^{*}(x)];{\lambda }^{*}(x)\}}^2$

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

Further relations:

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

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

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

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

$$\begin{array}{cccc}& a^6-f^6=2af+2a^5{f}^5& (a={\left[\frac{2{\lambda }^{*}(x)}{1-{\lambda }^{*}(x{)}^2}\right]}^{1/12})& (f={\left[\frac{2{\lambda }^{*}(25x)}{1-{\lambda }^{*}(25x{)}^2}\right]}^{1/12})\\ & a^8+b^8-7a^4{b}^4=2\sqrt{2}ab+2\sqrt{2}{a}^7{b}^7& (a={\left[\frac{2{\lambda }^{*}(x)}{1-{\lambda }^{*}(x{)}^2}\right]}^{1/12})& (b={\left[\frac{2{\lambda }^{*}(49x)}{1-{\lambda }^{*}(49x{)}^2}\right]}^{1/12})\\ & a^{12}-c^{12}=2\sqrt{2}(ac+a^3{c}^3)(1+3a^2{c}^2+a^4{c}^4)(2+3a^2{c}^2+2a^4{c}^4)& (a={\left[\frac{2{\lambda }^{*}(x)}{1-{\lambda }^{*}(x{)}^2}\right]}^{1/12})& (c={\left[\frac{2{\lambda }^{*}(121x)}{1-{\lambda }^{*}(121x{)}^2}\right]}^{1/12})\\ & (a^2-d^2)(a^4+d^4-7a^2{d}^2)[(a^2-d^2{)}^4-a^2{d}^2(a^2+d^2{)}^2]=8ad+8a^{13}{d}^{13}& (a={\left[\frac{2{\lambda }^{*}(x)}{1-{\lambda }^{*}(x{)}^2}\right]}^{1/12})& (d={\left[\frac{2{\lambda }^{*}(169x)}{1-{\lambda }^{*}(169x{)}^2}\right]}^{1/12})\end{array}$$

Special values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Lambda-star values of rational fractions:

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

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

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

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

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

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

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

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

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

Ramanujan's class invariants

Ramanujan's class invariants $G_n$ and $g_n$ are defined as

$G_n=2^{-1/4}{e}^{\pi \sqrt{n}/24}\prod _{k=0}^{\mathrm{\infty }}(1+e^{-(2k+1)\pi \sqrt{n}}),$

$g_n=2^{-1/4}{e}^{\pi \sqrt{n}/24}\prod _{k=0}^{\mathrm{\infty }}(1-e^{-(2k+1)\pi \sqrt{n}}),$

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

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

Identities with the class invariants include

$G_n=G_{1/n},g_n=\frac{1}{g_{4/n}},g_{4n}=2^{1/4}{g}_n{G}_n.$

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

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

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

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

Other appearances

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 $q^{-1}+20q-62q^3+\dots$, (sequence A007248 in the OEIS) 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.

References

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.

External links

• Modular lambda function at Fungrim