Modular lambda function

In mathematics, the elliptic 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 ${\displaystyle \mathbb {C} /\langle 1,\tau \rangle }$, 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 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 ${\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}$, and it is in fact Klein's modular j-invariant.

Modular properties

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 elliptic functions

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

${\displaystyle \lambda (\tau )={\Bigg (}{\frac {{\sqrt {2}}\,\eta ({\tfrac {\tau }{2}})\eta ^{2}(2\tau )}{\eta ^{3}(\tau )}}{\Bigg )}^{8}={\frac {16}{\left({\frac {\eta (\tau /2)}{\eta (2\tau )}}\right)^{8}+16}}={\frac {\theta _{2}^{4}(0,\tau )}{\theta _{3}^{4}(0,\tau )}}}$ 

and,

${\displaystyle {\frac {1}{{\big (}\lambda (\tau ){\big )}^{1/4}}}-{\big (}\lambda (\tau ){\big )}^{1/4}={\frac {1}{2}}\left({\frac {\eta ({\tfrac {\tau }{4}})}{\eta (\tau )}}\right)^{4}=2\,{\frac {\theta _{4}^{2}(0,{\tfrac {\tau }{2}})}{\theta _{2}^{2}(0,{\tfrac {\tau }{2}})}}}$ 

where^[5] for the nome ${\displaystyle q=e^{\pi i\tau }}$ ,

${\displaystyle \theta _{2}(0,\tau )=\sum _{n=-\infty }^{\infty }q^{\left({n+{\frac {1}{2}}}\right)^{2}}}$ 

${\displaystyle \theta _{3}(0,\tau )=\sum _{n=-\infty }^{\infty }q^{n^{2}}}$ 

${\displaystyle \theta _{4}(0,\tau )=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}}$ 

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

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

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 λ 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 ${\displaystyle y^{2}=x(x-1)(x-\lambda )}$ 

Elliptic modulus

Definition and computation of lambda-star

The function λ*(x) 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:

${\displaystyle K({\sqrt {1-k^{2}}})/K(k)={\sqrt {x}}}$ 

${\displaystyle k=\lambda ^{*}(x)}$ 

The values of λ*(x) can be computed as follows:

${\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{2}^{2}[0;\exp(-\pi {\sqrt {x}})]}{\vartheta _{3}^{2}[0;\exp(-\pi {\sqrt {x}})]}}}$ 

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

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

The functions λ* and λ are related to each other in this way:

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

Properties of lambda-star

Every λ*-value of a positive rational number is a positive algebraic number:

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

Elliptic integrals of the first and second kind of these special λ*-values are called elliptic integral singular values. They all can be expressed by polynomials of the gamma function, as Selberg and Chowla proved in 1967.

Following expression is valid for all n ∈ ℕ:

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

In this formula, dn is the Jacobi elliptic function delta amplitudinis.

By knowing one λ*-value, this formula can be used to computate related λ*-values:

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

In that formula, sn is the Jacobi elliptic function sinus amplitudinis. That formula works for all natural numbers.

Further relations:

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

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

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

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

${\displaystyle \left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/2}-\left[{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}\right]^{1/2}=2\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\left[{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}\right]^{1/12}+2\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{5/12}\left[{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}\right]^{5/12}}$ 

These are the relations between lambda-star and the Ramanujan-G-function:

${\displaystyle G(x)=\sin[2\arcsin[\lambda ^{*}(x)]]^{-1/12}=1/[{\sqrt[{12}]{2\lambda ^{*}(x)}}{\sqrt[{24}]{1-\lambda ^{*}(x)^{2}}}]}$ 

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

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

Special Values

Lambda-star-values of integer numbers:

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle \lambda ^{*}(14)=[2{\sqrt {2}}+2-({\sqrt {2}}+1)^{2}{\sqrt {4{\sqrt {2}}-5}}][({\sqrt {2}}+1)^{2}{\sqrt {4{\sqrt {2}}-5}}-{\sqrt {8{\sqrt {2}}+10}}]}$ 

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

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

${\displaystyle \lambda ^{*}(17)={\frac {1}{8{\sqrt {2}}}}({\sqrt {3{\sqrt {17}}+11}}-{\sqrt {5+{\sqrt {17}}}}-{\sqrt {2{\sqrt {2{\sqrt {17}}+2}}-4}})^{2}}$ 

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

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

${\displaystyle \lambda ^{*}(20)=({\sqrt {10}}-3)({\sqrt {5}}+2)({\sqrt {2}}-1)({\sqrt {{\sqrt {5}}-1}}-1)^{2}}$ 

${\displaystyle \lambda ^{*}(21)={\frac {1}{4{\sqrt {2}}}}({\sqrt {7}}-{\sqrt {3}})[({\sqrt {3}}+1){\sqrt {2{\sqrt {7}}-4}}-4+{\sqrt {7}}-{\sqrt {3}}]}$ 

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

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

${\displaystyle \lambda ^{*}(24)=(2+{\sqrt {3}})^{2}({\sqrt {3}}+{\sqrt {2}})[{\sqrt {{\sqrt {3}}+{\sqrt {2}}}}-({\sqrt {3}}-1)({\sqrt {2}}+1)]^{2}}$ 

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

Lambda-star-values of rational fractions:

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

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

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

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

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

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

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

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

${\displaystyle \lambda ^{*}({\frac {4}{5}})=({\sqrt {10}}+3)({\sqrt {5}}+2)({\sqrt {2}}+1)({\sqrt {{\sqrt {5}}-1}}-1)^{2}}$ 

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.^[8] 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.^[9]

Moonshine

The function ${\displaystyle {\frac {16}{\lambda (2\tau )}}-8}$  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.

Footnotes

1. Chandrasekharan (1985) p.115
2. Chandrasekharan (1985) p.109
3. Chandrasekharan (1985) p.110
4. Chandrasekharan (1985) p.108
5. Chandrasekharan (1985) p.63
6. Chandrasekharan (1985) p.117
7. Rankin (1977) pp.226–228
8. Chandrasekharan (1985) p.121
9. Chandrasekharan (1985) p.118

References

• 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.