Modular lambda function

In mathematics, the modular lambda function λ(τ)[note 1] 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

, where the map is defined as the quotient by the [−1] involution.

is the nome, is given by: 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

, and it is in fact Klein's modular j-invariant.

is that of the anharmonic group, giving the six values of the cross-ratio:[3] It is the square of the elliptic modulus,[4] that is,

In terms of the Dedekind eta function

we have[4] Since the three half-period values are distinct, this shows that

is the complete elliptic integral of the first kind with parameter

Then The modular equation of degree

is a prime number) is an algebraic equation in

) can be thought of as a holomorphic function on the upper half-plane

, the modular equations can be used to give algebraic values of

) gives the value of the elliptic modulus

, for which the complete elliptic integral of the first kind

are related by following expression: The values of

value of a positive rational number is a positive algebraic number:

(the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any

is the Jacobi elliptic function delta amplitudinis with modulus

value, this formula can be used to compute related

is the Jacobi elliptic function sinus amplitudinis with modulus

Lambda-star values of integer numbers of 4n-3-type: Lambda-star values of integer numbers of 4n-2-type: Lambda-star values of integer numbers of 4n-1-type: Lambda-star values of integer numbers of 4n-type: Lambda-star values of rational fractions: Ramanujan's class invariants

, the class invariants are algebraic numbers.

For example Identities with the class invariants include[14] The class invariants are very closely related to the Weber modular functions

These are the relations between lambda-star and the class invariants: 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.

[15] Suppose if possible that f is entire and does not take the values 0 and 1.

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.

, is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.