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 $\mathbb{ℂ}/⟨1,\tau ⟩$, 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 $S{L}_2(\mathbb{\mathbb{Z}})$, 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:

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

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,

$\lambda (\tau )=(\frac{\sqrt{2}\eta (\frac{\tau }{2}){\eta }^2(2\tau )}{{\eta }^3(\tau )}{)}^8=\frac{16}{{\left(\frac{\eta (\tau /2)}{\eta (2\tau )}\right)}^8+16}=\frac{{\theta }_2^4(0,\tau )}{{\theta }_3^4(0,\tau )}$

and,

$\frac{1}{(\lambda (\tau ){)}^{1/4}}-(\lambda (\tau ){)}^{1/4}=\frac{1}{2}{\left(\frac{\eta (\frac{\tau }{4})}{\eta (\tau )}\right)}^4=2\frac{{\theta }_4^2(0,\frac{\tau }{2})}{{\theta }_2^2(0,\frac{\tau }{2})}$

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

${\theta }_2(0,\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{{\left(n+\frac{1}{2}\right)}^2}$

${\theta }_3(0,\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{n^2}$

${\theta }_4(0,\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{n^2}$

In terms of the half-periods of Weierstrass's elliptic functions, let $[{\omega }_1,{\omega }_2]$ be a fundamental pair of periods with $\tau =\frac{{\omega }_2}{{\omega }_1}$.

$e_1=\mathrm{\wp }\left(\frac{{\omega }_1}{2}\right),e_2=\mathrm{\wp }\left(\frac{{\omega }_2}{2}\right),e_3=\mathrm{\wp }\left(\frac{{\omega }_1+{\omega }_2}{2}\right)$

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 ${\mathrm{\Gamma }}_0(4)$, 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