# 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 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 $\mathbb {C} /\langle 1,\tau \rangle$, 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:
    $\lambda (\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots$. OEIS: A115977
By symmetrizing the lambda function under the canonical action of the symmetric group S$_3$ 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,
    $(1+u^{4})^{2}v^{8}-4u^{4}=0,$
    $u^{4}-v^{4}+2uv(1-u^{2}v^{2})=0,$
    $u^{6}-v^{6}+5u^{2}v^{2}(u^{2}-v^{2})+4uv(1-u^{4}v^{4})=0,$
    $(1-u^{8})(1-v^{8})-(1-uv)^{8}=0.$
The quantity $v$ (and hence $u$) can be thought of as a holomorphic function on the upper half-plane $\operatorname {Im} \tau >0$:
    ${\begin{aligned}v&=\prod _{k=1}^{\infty }\tanh {\frac {(k-1/2)\pi i}{\tau }}={\sqrt {2}}e^{\pi i\tau /8}{\frac {\sum _{k\in \mathbb {Z} }e^{(2k^{2}+k)\pi i\tau }}{\sum _{k\in \mathbb {Z} }e^{k^{2}\pi i\tau }}}\\&={\cfrac {{\sqrt {2}}e^{\pi i\tau /8}}{1+{\cfrac {e^{\pi i\tau }}{1+e^{\pi i\tau }+{\cfrac {e^{2\pi i\tau }}{1+e^{2\pi i\tau }+{\cfrac {e^{3\pi i\tau }}{1+e^{3\pi i\tau }+\ddots }}}}}}}}\end{aligned}}$
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.

## Lambda-star
### 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:
    $\lambda ^{*}(x)={\frac {\theta _{2}^{2}(i{\sqrt {x}})}{\theta _{3}^{2}(i{\sqrt {x}})}}$
    $\lambda ^{*}(x)=\left[\sum _{a=-\infty }^{\infty }\exp[-(a+1/2)^{2}\pi {\sqrt {x}}]\right]^{2}\left[\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x}})\right]^{-2}$
    $\lambda ^{*}(x)=\left[\sum _{a=-\infty }^{\infty }\operatorname {sech} [(a+1/2)\pi {\sqrt {x}}]\right]\left[\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {x}})\right]^{-1}$
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:
    $\lambda ^{*}(n^{2}x)=\lambda ^{*}(x)^{n}\prod _{a=1}^{n}\operatorname {sn} \left\{{\frac {2a-1}{n}}K[\lambda ^{*}(x)];\lambda ^{*}(x)\right\}^{2}$
where $n\in \mathbb {N}$ and $\operatorname {sn}$ is the Jacobi elliptic function sinus amplitudinis with modulus $k$.
Further relations:
    $\lambda ^{*}(x)^{2}+\lambda ^{*}(1/x)^{2}=1$
    $[\lambda ^{*}(x)+1][\lambda ^{*}(4/x)+1]=2$
    $\lambda ^{*}(4x)={\frac {1-{\sqrt {1-\lambda ^{*}(x)^{2}}}}{1+{\sqrt {1-\lambda ^{*}(x)^{2}}}}}=\tan \left\{{\frac {1}{2}}\arcsin[\lambda ^{*}(x)]\right\}^{2}$
    $\lambda ^{*}(x)-\lambda ^{*}(9x)=2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{1/4}-2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{3/4}$
${\begin{aligned}&a^{6}-f^{6}=2af+2a^{5}f^{5}\,&\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)&\left(f=\left[{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}\right]^{1/12}\right)\\&a^{8}+b^{8}-7a^{4}b^{4}=2{\sqrt {2}}ab+2{\sqrt {2}}a^{7}b^{7}\,&\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)&\left(b=\left[{\frac {2\lambda ^{*}(49x)}{1-\lambda ^{*}(49x)^{2}}}\right]^{1/12}\right)\\&a^{12}-c^{12}=2{\sqrt {2}}(ac+a^{3}c^{3})(1+3a^{2}c^{2}+a^{4}c^{4})(2+3a^{2}c^{2}+2a^{4}c^{4})\,&\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)&\left(c=\left[{\frac {2\lambda ^{*}(121x)}{1-\lambda ^{*}(121x)^{2}}}\right]^{1/12}\right)\\&(a^{2}-d^{2})(a^{4}+d^{4}-7a^{2}d^{2})[(a^{2}-d^{2})^{4}-a^{2}d^{2}(a^{2}+d^{2})^{2}]=8ad+8a^{13}d^{13}\,&\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)&\left(d=\left[{\frac {2\lambda ^{*}(169x)}{1-\lambda ^{*}(169x)^{2}}}\right]^{1/12}\right)\end{aligned}}$

### 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
    $G_{n}=G_{1/n},\quad g_{n}={\frac {1}{g_{4/n}}},\quad g_{4n}=2^{1/4}g_{n}G_{n}.$
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:
    $G_{n}=\sin\{2\arcsin[\lambda ^{*}(n)]\}^{-1/12}=1{\Big /}\left[{\sqrt[{12}]{2\lambda ^{*}(n)}}{\sqrt[{24}]{1-\lambda ^{*}(n)^{2}}}\right]$
    $g_{n}=\tan\{2\arctan[\lambda ^{*}(n)]\}^{-1/12}={\sqrt[{12}]{[1-\lambda ^{*}(n)^{2}]/[2\lambda ^{*}(n)]}}$
    $\lambda ^{*}(n)=\tan \left\{{\frac {1}{2}}\arctan[g_{n}^{-12}]\right\}={\sqrt {g_{n}^{24}+1}}-g_{n}^{12}$

## 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 $q^{-1}+20q-62q^{3}+\dots$, OEIS: A007248 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.

## Footnotes

* ↑ Chandrasekharan (1985) p.115
* ↑ Chandrasekharan (1985) p.109
* ↑ Chandrasekharan (1985) p.110
* 1 2 3 4 Chandrasekharan (1985) p.108
* ↑ Chandrasekharan (1985) p.63
* ↑ Chandrasekharan (1985) p.117
* ↑ Rankin (1977) pp.226–228
* ↑ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 103–109, 134
* 1 2 Jacobi, Carl Gustav Jacob (1829). Fundamenta nova theoriae functionum ellipticarum (in Latin). p. 42
* ↑ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 152
* ↑ Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences. 35 (7): 373. doi:10.1073/PNAS.35.7.371. PMC 1063041. S2CID 45071481.
* ↑ Chowla, S.; Selberg, A. "On Epstein's Zeta-Function". EuDML. pp. 86–110.
* ↑ Berndt, Bruce C.; Chan, Heng Huat; Zhang, Liang-Cheng (6 June 1997). "Ramanujan's class invariants, Kronecker's limit formula, and modular equations". Transactions of the American Mathematical Society. 349 (6): 2125–2173.
* ↑ Eymard, Pierre; Lafon, Jean-Pierre (1999). Autour du nombre Pi (in French). HERMANN. ISBN 2705614435. p. 240
* ↑ Chandrasekharan (1985) p.121
* ↑ Chandrasekharan (1985) p.118