Modular lambda function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the modular lambda function λ(τ) is a highly symmetric <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a> on the complex <a href="/facts/Upper_half-plane/7fPqrowr">upper half-plane</a>. It is invariant under the fractional linear action of the <a href="/facts/Congruence_subgroup/V334rHwz">congruence group</a> Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the <a href="/facts/Modular_curve/UE3kT5SW">modular curve</a> X(2). Over any point τ, its value can be described as a <a href="/facts/Cross_ratio/tj4DK3Vk">cross ratio</a> of the branch points of a ramified double cover of the projective line by the <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curve</a> 
 
 
 
 
 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 <a href="/facts/Nome_(mathematics)/C3yIM7eR">nome</a>, 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 }
 
. <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A115977
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 <a href="/facts/J-invariant/H1a38AJF">j-invariant</a>.

Modular lambda function open-in-new

Modular lambda function