Modular lambda function

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In mathematics, the modular lambda function λ(τ)^[1] is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence subgroup ⁠ ${\displaystyle \Gamma (2)}$ ⁠ of ${\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}$ , and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve ⁠ ${\displaystyle X(2)}$ ⁠. Over any point ⁠ ${\displaystyle \tau }$ ⁠, 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

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

(sequence A115977 in the OEIS). By symmetrizing the lambda function under the canonical action of the symmetric group ⁠ ${\displaystyle X(2)}$ ⁠ on ⁠ ${\displaystyle S_{3$ ⁠, 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^[2]

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

The generators of the modular group act by^[3]

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

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

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:^[4]

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

Relations to other functions#edit

It is the square of the elliptic modulus,^[5] that is, ${\displaystyle \lambda (\tau )=k^{2}(\tau )}$ . In terms of the Dedekind eta function ${\displaystyle \eta (\tau )}$ and theta functions,^[5]

${\displaystyle \lambda (\tau )={\Bigg (}{\frac \sqrt {2\,\eta ({\tfrac {\tau }{2)\eta ^{2}(2\tau )}{\eta ^{3}(\tau ){\Bigg )}^{8}={\frac {16}{\left({\frac {\eta (\tau$

and,

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

where^[6]

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

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

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

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^[5]

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

Since the three half-period values are distinct, this shows that ${\displaystyle \lambda }$ does not take the value 0 or 1.^[5]

The relation to the j-invariant is^[7]^[8]

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

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,^[9]

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

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

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

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

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

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

Since ${\displaystyle \lambda (i)=1/2}$ , the modular equations can be used to give algebraic values of ${\displaystyle \lambda (pi)}$ for any prime ${\displaystyle p}$ .^[10] The algebraic values of ${\displaystyle \lambda (ni)}$ are also given by^[11]^[12]

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

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

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 ${\displaystyle \lambda ^{*}(x)}$ ^[13] (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:

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

${\displaystyle \lambda ^{*}(x)=\left[\sum _{a=-\infty }^{\infty }\exp[-(a+1/2)^{2}\pi {\sqrt {x]\right]^{2}\left[\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt &#123$

${\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 {\$

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:

${\displaystyle \lambda ^{*}(x)\in {\overline {\mathbb {Q} _{+}\quad \forall x\in \mathbb {Q} ^{+}.}$

{{mathimg|8e8a431d9b700eb724438607c48e13519e64fe9f|width=9.424ex|height=2.843ex|va=-0.838ex|alt={\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.^[14]^[15]

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

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

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:^[11]

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

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

Further relations:

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

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

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

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

{{mathimg|c2fca17619134acaa8774d9c38c77bb968fa4f8e|width=133.202ex|height=30.843ex|va=-14.838ex|alt={\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[&#1}}

Ramanujan's class invariants#edit

Ramanujan's class invariants ${\displaystyle G_{n$ and ${\displaystyle g_{n$ are defined as^[16]

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

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

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^[17]

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

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:

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

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

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

Other appearances#edit

Little Picard theorem#edit

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.^[18] Suppose if possible that ${\displaystyle f}$ is entire and does not take the values 0 and 1. Since ${\displaystyle \lambda }$ is holomorphic, it has a local holomorphic inverse ${\displaystyle \omega }$ defined away from ${\displaystyle 0,1,\infty }$ . Consider the function ${\displaystyle z\mapsto \omega (f(z))}$ . By the Monodromy theorem this is holomorphic and maps the complex plane ${\displaystyle \mathbb {C} }$ to the upper half plane. From this it is easy to construct a holomorphic function from ${\displaystyle \mathbb {C} }$ to the unit disc, which by Liouville's theorem must be constant.^[19]

Moonshine#edit

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

${\displaystyle q^{-1}+20q-62q^{3}+\dots }$

(sequence A007248 in the OEIS), 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.

Footnotes#edit

^ ${\displaystyle \lambda (\tau )}$ is not a modular function (per the Wikipedia definition), but every modular function is a rational function in ${\displaystyle \lambda (\tau )}$ . Some authors use a non-equivalent definition of "modular functions".
^ Chandrasekharan (1985), p. 115.
^ Chandrasekharan (1985), p. 109.
^ Chandrasekharan (1985), p. 110.
^ a b c d 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
^ For any prime power, we can iterate the modular equation of degree ${\displaystyle p}$ . This process can be used to give algebraic values of ${\displaystyle \lambda (ni)}$ for any ${\displaystyle n\in \mathbb {N} .}$
^ a b Jacobi, Carl Gustav Jacob (1829). Fundamenta nova theoriae functionum ellipticarum (in Latin). p. 42
^ ${\displaystyle \operatorname {sl} a\varpi }$ is algebraic for every ${\displaystyle a\in \mathbb {Q} .}$
^ 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.

References#edit

Other#edit

Abramowitz, Milton; Stegun, Irene A., eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, ISBN 978-0-486-61272-0, Zbl 0543.33001
Chandrasekharan, K. (1985), Elliptic Functions, Grundlehren der mathematischen Wissenschaften, vol. 281, Springer-Verlag, pp. 108–121, ISBN 3-540-15295-4, Zbl 0575.33001
Conway, John Horton; Norton, Simon (1979), "Monstrous moonshine", Bulletin of the London Mathematical Society, 11 (3): 308–339, doi:10.1112/blms/11.3.308, MR 0554399, Zbl 0424.20010
Rankin, Robert A. (1977), Modular Forms and Functions, Cambridge University Press, ISBN 0-521-21212-X, Zbl 0376.10020
Reinhardt, W. P.; Walker, P. L. (2010), "Elliptic Modular Function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

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#edit

Modular lambda function at Fungrim

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