Modular lambda function

The function ${\displaystyle \lambda (\tau )}$  is invariant under the group generated by^[1]

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

The generators of the modular group act by^[2]

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

${\displaystyle \tau \mapsto -{\frac {1}{\tau }}\ :\ \lambda \mapsto 1-\lambda \ .}$ 

Consequently, the action of the modular group on ${\displaystyle \lambda (\tau )}$  is that of the anharmonic group, giving the six values of the cross-ratio:^[3]

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

It is the square of the elliptic modulus,^[4] that is, ${\displaystyle \lambda (\tau )=k^{2}(\tau )}$ . In terms of the Dedekind eta function ${\displaystyle \eta (\tau )}$  and theta functions,^[4]

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

and,

${\displaystyle {\frac {1}{{\big (}\lambda (\tau ){\big )}^{1/4}}}-{\big (}\lambda (\tau ){\big )}^{1/4}={\frac {1}{2}}\left({\frac {\eta ({\tfrac {\tau }{4}})}{\eta (\tau )}}\right)^{4}=2\,{\frac {\theta _{4}^{2}({\tfrac {\tau }{2}})}{\theta _{2}^{2}({\tfrac {\tau }{2}})}}}$ 

where^[5]

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

${\displaystyle \theta _{3}(\tau )=\sum _{n=-\infty }^{\infty }e^{\pi i\tau n^{2}}}$ 

${\displaystyle \theta _{4}(\tau )=\sum _{n=-\infty }^{\infty }(-1)^{n}e^{\pi i\tau n^{2}}}$ 

In terms of the half-periods of Weierstrass's elliptic functions, let ${\displaystyle [\omega _{1},\omega _{2}]}$  be a fundamental pair of periods with ${\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}$ .

${\displaystyle e_{1}=\wp \left({\frac {\omega _{1}}{2}}\right),\quad e_{2}=\wp \left({\frac {\omega _{2}}{2}}\right),\quad e_{3}=\wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)}$ 

we have^[4]

${\displaystyle \lambda ={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}\,.}$ 

Since the three half-period values are distinct, this shows that ${\displaystyle \lambda }$  does not take the value 0 or 1.^[4]

The relation to the j-invariant is^[6][7]

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

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

Given ${\displaystyle m\in \mathbb {C} \setminus \{0,1\}}$ , let

${\displaystyle \tau =i{\frac {K\{1-m\}}{K\{m\}}}}$ 

where ${\displaystyle K}$  is the complete elliptic integral of the first kind with parameter ${\displaystyle m=k^{2}}$ . Then

${\displaystyle \lambda (\tau )=m.}$ 

The modular equation of degree ${\displaystyle p}$  (where ${\displaystyle p}$  is a prime number) is an algebraic equation in ${\displaystyle \lambda (p\tau )}$  and ${\displaystyle \lambda (\tau )}$ . If ${\displaystyle \lambda (p\tau )=u^{8}}$  and ${\displaystyle \lambda (\tau )=v^{8}}$ , the modular equations of degrees ${\displaystyle p=2,3,5,7}$  are, respectively,^[8]

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

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

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

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

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

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

Since ${\displaystyle \lambda (i)=1/2}$ , the modular equations can be used to give algebraic values of ${\displaystyle \lambda (pi)}$  for any prime ${\displaystyle p}$ .^{[note 2]} The algebraic values of ${\displaystyle \lambda (ni)}$  are also given by^{[9][note 3]}

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

${\displaystyle \lambda (ni)={\frac {1}{2^{n}}}\prod _{k=1}^{n-1}\left(1-\operatorname {sl} ^{2}{\frac {k\varpi }{n}}\right)^{2}\quad (n\,{\text{odd}})}$ 

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

The function ${\displaystyle \lambda ^{*}(x)}$ ^[10] (where ${\displaystyle x\in \mathbb {R} ^{+}}$ ) gives the value of the elliptic modulus ${\displaystyle k}$ , for which the complete elliptic integral of the first kind ${\displaystyle K(k)}$  and its complementary counterpart ${\displaystyle K({\sqrt {1-k^{2}}})}$  are related by following expression:

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

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

${\displaystyle \lambda ^{*}(x)={\frac {\theta _{2}^{2}(i{\sqrt {x}})}{\theta _{3}^{2}(i{\sqrt {x}})}}}$ 

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

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

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

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

Every ${\displaystyle \lambda ^{*}}$  value of a positive rational number is a positive algebraic number:

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

${\displaystyle K(\lambda ^{*}(x))}$  and ${\displaystyle E(\lambda ^{*}(x))}$  (the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any ${\displaystyle x\in \mathbb {Q} ^{+}}$ , as Selberg and Chowla proved in 1949.^[11][12]

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

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

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

By knowing one ${\displaystyle \lambda ^{*}}$  value, this formula can be used to compute related ${\displaystyle \lambda ^{*}}$  values:^[9]

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

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

Further relations:

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

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

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

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

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

Ramanujan's class invariants ${\displaystyle G_{n}}$  and ${\displaystyle g_{n}}$  are defined as^[13]

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

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

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

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

Identities with the class invariants include^[14]

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

The class invariants are very closely related to the Weber modular functions ${\displaystyle {\mathfrak {f}}}$  and ${\displaystyle {\mathfrak {f}}_{1}}$ . These are the relations between lambda-star and the class invariants:

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

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

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

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.^[15] 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.^[16]

The function ${\displaystyle \tau \mapsto 16/\lambda (2\tau )-8}$  is the normalized Hauptmodul for the group ${\displaystyle \Gamma _{0}(4)}$ , and its q-expansion ${\displaystyle q^{-1}+20q-62q^{3}+\dots }$ , OEIS: A007248 where ${\displaystyle q=e^{2\pi i\tau }}$ , is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

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