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 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 ⟩ {\displaystyle \mathbb {C} /\langle 1,\tau \rangle } , where the map is defined as the quotient by the [−1] involution. The q-expansion, where q = e π i τ {\displaystyle q=e^{\pi i\tau }} 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 + … {\displaystyle \lambda (\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots } . (sequence A115977 in the OEIS) 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 SL 2 ( Z ) {\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )} , and it is in fact Klein's modular j-invariant.
Modular properties The function λ ( τ ) {\displaystyle \lambda (\tau )} is invariant under the group generated by
τ ↦ τ + 2 ; τ ↦ τ 1 − 2 τ . {\displaystyle \tau \mapsto \tau +2\ ;\ \tau \mapsto {\frac {\tau }{1-2\tau }}\ .}
The generators of the modular group act by
τ ↦ τ + 1 : λ ↦ λ λ − 1 ; {\displaystyle \tau \mapsto \tau +1\ :\ \lambda \mapsto {\frac {\lambda }{\lambda -1}}\,;}
τ ↦ − 1 τ : λ ↦ 1 − λ . {\displaystyle \tau \mapsto -{\frac {1}{\tau }}\ :\ \lambda \mapsto 1-\lambda \ .}
Consequently, the action of the modular group on λ ( τ ) {\displaystyle \lambda (\tau )} 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 } . {\displaystyle \left\lbrace {\lambda ,{\frac {1}{1-\lambda }},{\frac {\lambda -1}{\lambda }},{\frac {1}{\lambda }},{\frac {\lambda }{\lambda -1}},1-\lambda }\right\rbrace \ .}
Relations to other functions It is the square of the elliptic modulus, that is, λ ( τ ) = k 2 ( τ ) {\displaystyle \lambda (\tau )=k^{2}(\tau )} . In terms of the Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} and theta functions,
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