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In mathematics, the 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 C / ⟨ 1 , τ ⟩ {\displaystyle \mathbb {C} /\langle 1,\tau \rangle } , where the map is defined as the quotient by the involution.
The q-expansion, where q = e π i τ {\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, and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group SL 2 {\displaystyle \operatorname {SL} _{2}} , and it is in fact Klein's modular j-invariant.