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Modular lambda function

Symmetric holomorphic function

Inmathematics, the modular lambda function λ(τ)^{[note 1]} 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 ${\displaystyle \mathbb {C} /\langle 1,\tau \rangle }$, where the map is defined as the quotient by the [−1] involution.

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

${\displaystyle \lambda (\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots }$. OEIS: A115977

By symmetrizing the lambda function under the canonical action of the symmetric group S₃onX(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group ${\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}$, and it is in fact Klein's modular j-invariant.

Modular properties[edit]

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]

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

Relations to other functions[edit]

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-invariantis^[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

${\displaystyle \lambda (\tau )=m.}$

Modular equations[edit]

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 valuesof${\displaystyle \lambda ($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.

Lambda-star[edit]

Definition and computation of lambda-star[edit]

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}})}}}

Properties of lambda-star[edit]

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}}