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 ${\displaystyle \mathbb {C} /\langle 1,\tau \rangle }$, where the map is defined as the quotient by the [−1] involution.

The q-expansion, where ${\displaystyle q=e^{\pi i\tau }}$ 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 ${\displaystyle SL_{2}(\mathbb {Z} )}$, and it is in fact Klein's modular j-invariant.

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