Modular lambda function

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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. OEISA115977

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[edit]

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 -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[edit]

Other elliptic functions[edit]

It is the square of the Jacobi modulus,[4] that is, λ(τ) = k2(τ). 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{1}{2}\tau)\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 [ω1, ω2] be a fundamental pair of periods with τ = ω21.

 e_1 = \wp(\omega_1/2), e_2 = \wp(\omega_2/2), e_3 = \wp((\omega_1+\omega_2)/2)

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) = 256 \frac{(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[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.[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[edit]

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[edit]

  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