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:

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

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[1]

$\tau \mapsto \tau+2 \ ;\ \tau \mapsto \frac{\tau}{1-2\tau} \ .$

The generators of the modular group act by[2]

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

Consequently, the action of the modular group on $\lambda(\tau)$ is that of the anharmonic group, giving the six values of the cross-ratio:[3]

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

Other appearances

Other elliptic functions

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

$\lambda(\tau) = \frac{\theta_2^4(0,\tau)}{\theta_3^4(0,\tau)} = \left[\frac{\sqrt{2}\,\eta(\tfrac{\tau}{2})\eta^2(2\tau)}{\eta^3(\tau)}\right]^8$

where[5]

$\theta_2(0,\tau) = \sum_{n=-\infty}^\infty q^{\left({n+\frac12}\right)^2} \mathrm{ and } \ \theta_3(0,\tau) = \sum_{n=-\infty}^\infty q^{n^2}$

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

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[4]

$\lambda = \frac{e_3-e_2}{e_1-e_2} \, .$

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]

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

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.[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 $\frac{16}{\lambda(2\tau)} - 8$ is the normalized Hauptmodul for the group $\Gamma_0(4)$, and its q-expansion $q^{-1} + 20q - 62q^3 + \dots$ is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

References

1. ^ Chandrasekharan (1985) p.115
2. ^ Chandrasekharan (1985) p.109
3. ^ Chandrasekharan (1985) p.110
4. ^ a b c d 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