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 $\mathbb {C} /\langle 1,\tau \rangle$, 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 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 $SL_{2}(\mathbb {Z} )$, 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:

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

Other appearances

Other elliptic functions

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

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

and,

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

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

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

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

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

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

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

we have

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

Since the three half-period values are distinct, this shows that λ 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 $y^{2}=x(x-1)(x-\lambda )$

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

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, 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