Harman Patil (Editor)

Modular lambda function

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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 C / ⟨ 1 , τ ⟩ , where the map is defined as the quotient by the [−1] involution.

Contents

The q-expansion, where q = e π i τ is the nome, is given by:

λ ( τ ) = 16 q − 128 q 2 + 704 q 3 − 3072 q 4 + 11488 q 5 − 38400 q 6 + … .  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 ( Z ) , and it is in fact Klein's modular j-invariant.

Modular properties

The function λ ( τ ) is invariant under the group generated by

τ ↦ τ + 2   ;   τ ↦ τ 1 − 2 τ   .

The generators of the modular group act by

τ ↦ τ + 1   :   λ ↦ λ λ − 1 ; τ ↦ − 1 τ   :   λ ↦ 1 − λ   .

Consequently, the action of the modular group on λ ( τ ) is that of the anharmonic group, giving the six values of the cross-ratio:

{ λ , 1 1 − λ , λ − 1 λ , 1 λ , λ λ − 1 , 1 − λ }   .

Other elliptic functions

It is the square of the Jacobi modulus, that is, λ ( τ ) = k 2 ( τ ) . In terms of the Dedekind eta function η ( τ ) and theta functions,

λ ( τ ) = ( 2 η ( τ 2 ) η 2 ( 2 τ ) η 3 ( τ ) ) 8 = θ 2 4 ( 0 , τ ) θ 3 4 ( 0 , τ )

and,

1 ( λ ( τ ) ) 1 / 4 − ( λ ( τ ) ) 1 / 4 = 1 2 ( η ( τ 4 ) η ( τ ) ) 4 = 2 θ 4 2 ( 0 , τ 2 ) θ 2 2 ( 0 , τ 2 )

where for the nome q = e π i τ ,

θ 2 ( 0 , τ ) = ∑ n = − ∞ ∞ q ( n + 1 2 ) 2 θ 3 ( 0 , τ ) = ∑ n = − ∞ ∞ q n 2 θ 4 ( 0 , τ ) = ∑ n = − ∞ ∞ ( − 1 ) n q n 2

In terms of the half-periods of Weierstrass's elliptic functions, let [ ω 1 , ω 2 ] be a fundamental pair of periods with τ = ω 2 ω 1 .

e 1 = ℘ ( ω 1 2 ) , e 2 = ℘ ( ω 2 2 ) , e 3 = ℘ ( ω 1 + ω 2 2 )

we have

λ = 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 ( τ ) = 256 ( 1 − λ + λ 2 ) 3 λ 2 ( 1 − λ ) 2   .

which is the j-invariant of the elliptic curve of Legendre form y 2 = x ( x − 1 ) ( x − λ )

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 16 λ ( 2 τ ) − 8 is the normalized Hauptmodul for the group Γ 0 ( 4 ) , and its q-expansion q − 1 + 20 q − 62 q 3 + … is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

References

Modular lambda function Wikipedia
(Text) CC BY-SA