In mathematics, the 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 groupΓ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curveX(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.
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.
The function is invariant under the group generated by
Elliptic integrals of the first and second kind of these special λ*-values are called elliptic integral singular values. They all can be expressed by polynomials of the gamma function, as Selberg and Chowla proved in 1967.
By knowing one λ*-value, this formula can be used to compute related λ*-values:
In that formula, sn is the Jacobi elliptic function sinus amplitudinis. That formula works for all natural numbers.
Ramanujan's class invariants
Ramanujan's class invariants and are defined as
These are the relations between lambda-star and Ramanujan's class invariants:
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:
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.
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