Modular lambda function


In mathematics, the elliptic modular lambda function λ 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, and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X. 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, where the map is defined as the quotient by the involution.
The q-expansion, where is the nome, is given by:
By symmetrizing the lambda function under the canonical action of the symmetric group S3 on X, and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group, and it is in fact Klein's modular j-invariant.

Modular properties

The function is invariant under the group generated by
The generators of the modular group act by
Consequently, the action of the modular group on is that of the anharmonic group, giving the six values of the cross-ratio:

Other appearances

Other elliptic functions

It is the square of the Jacobi modulus, that is,. In terms of the Dedekind eta function and theta functions,
and,
where for the nome,
In terms of the half-periods of Weierstrass's elliptic functions, let be a fundamental pair of periods with.
we have
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
which is the j-invariant of the elliptic curve of Legendre form

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. 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 is the normalized Hauptmodul for the group, and its q-expansion, where, is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

Footnotes