Modular lambda function

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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 , where the map is defined as the quotient by the [−1] involution.

The q-expansion, where is the nome, is given by:

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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.

Modular properties[edit]

The function is invariant under the group generated by[1]

The generators of the modular group act by[2]

Consequently, the action of the modular group on is that of the anharmonic group, giving the six values of the cross-ratio:[3]

Other appearances[edit]

Other elliptic functions[edit]

It is the square of the Jacobi modulus,[4] that is, . In terms of the Dedekind eta function and theta functions,[4]

and,

where[5] for the nome ,

In terms of the half-periods of Weierstrass's elliptic functions, let be a fundamental pair of periods with .

we have[4]

Since the three half-period values are distinct, this shows that λ does not take the value 0 or 1.[4]

The relation to the j-invariant is[6][7]

which is the j-invariant of the elliptic curve of Legendre form

Little Picard theorem[edit]

The lambda function is used in the original proof of the Little Picard theorem, that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.[8] Suppose if possible that f is entire and does not take the values 0 and 1. Since λ is holomorphic, it has a local holomorphic inverse ω defined away from 0,1,∞. Consider the function z → ω(f(z)). By the Monodromy theorem this is holomorphic and maps the complex plane C to the upper half plane. From this it is easy to construct a holomorphic function from C to the unit disc, which by Liouville's theorem must be constant.[9]

Moonshine[edit]

The function is the normalized Hauptmodul for the group , and its q-expansion is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

References[edit]

  1. ^ Chandrasekharan (1985) p.115
  2. ^ Chandrasekharan (1985) p.109
  3. ^ Chandrasekharan (1985) p.110
  4. ^ a b c d Chandrasekharan (1985) p.108
  5. ^ Chandrasekharan (1985) p.63
  6. ^ Chandrasekharan (1985) p.117
  7. ^ Rankin (1977) pp.226–228
  8. ^ Chandrasekharan (1985) p.121
  9. ^ Chandrasekharan (1985) p.118