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Concept
Modular lambda function
Summary
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 , where the map is defined as the quotient by the [−1] involution.
The q-expansion, where 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 , and it is in fact Klein's modular j-invariant.
The function is invariant under the group generated by
The generators of the modular group act by
Consequently, the action of the modular group on is that of the anharmonic group, giving the six values of the cross-ratio:
It is the square of the elliptic modulus, that is, . In terms of the Dedekind eta function and theta functions,
and,
where
In terms of the half-periods of Weierstrass's elliptic functions, let be a fundamental pair of periods with .
we have
Since the three half-period values are distinct, this shows that does not take the value 0 or 1.
The relation to the j-invariant is
which is the j-invariant of the elliptic curve of Legendre form
Given , let
where is the complete elliptic integral of the first kind with parameter .
Then
The modular equation of degree (where is a prime number) is an algebraic equation in and . If and , the modular equations of degrees are, respectively,
The quantity (and hence ) can be thought of as a holomorphic function on the upper half-plane :
Since , the modular equations can be used to give algebraic values of for any prime . The algebraic values of are also given by
where is the lemniscate sine and is the lemniscate constant.
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