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 is given by:

$\lambda(\tau) = 16q^{1/2} - 128q + 704 q^{3/2} - 3072q^2 + 11488q^{5/2} - 38400q^3 + \dots$.

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.

Other appearances

It is the square of the Jacobi modulus, i.e., $\lambda(\tau) = k^2 (\tau). \,$.

The function $\frac{16}{\lambda(2\tau)} - 8$ is the normalized Hauptmodul for the group Γ₀(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

• 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 
• 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, MR0554399 .

External links

• http://dlmf.nist.gov/23.15#E6
• http://www.research.att.com/~njas/sequences/A115977