Math 2.0

Number theory Object

Modular lambda function

Builds on Upper half-planeTheta functionCongruence subgroupj-invariantPicard's little theorem

Description

Let ℍ={τ∈ℂ:Im⁡τ>0}\mathbb{H} = \set{\tau \in \C : \operatorname{Im}\tau > 0} be the upper half-plane, and for τ∈ℍ\tau \in \mathbb{H} let q=eπiτq = e^{\pi i \tau}. Let θ2\theta_2 and θ3\theta_3 be the theta functions θ2(τ)=∑n∈ℤq(n+1/2)2,θ3(τ)=∑n∈ℤqn2.\theta_2(\tau) = \sum_{n \in \Z} q^{(n + 1/2)^2}, \qquad \theta_3(\tau) = \sum_{n \in \Z} q^{n^2}. The modular lambda function is λ(τ)=θ2(τ)4θ3(τ)4,τ∈ℍ.\lambda(\tau) = \frac{\theta_2(\tau)^4}{\theta_3(\tau)^4}, \qquad \tau \in \mathbb{H}.

  1. λ\lambda is holomorphic on ℍ\mathbb{H} and never takes the values 00 and 11.
  2. λ(τ+1)=λ(τ)λ(τ)−1\lambda(\tau + 1) = \dfrac{\lambda(\tau)}{\lambda(\tau) - 1} and λ(−1/τ)=1−λ(τ)\lambda(-1/\tau) = 1 - \lambda(\tau).
  3. λ\lambda is invariant under the principal congruence subgroup Γ(2)\Gamma(2) of level 22.
  4. λ(τ)=16q+O(q2)\lambda(\tau) = 16q + O(q^2) as Im⁡τ→∞\operatorname{Im}\tau \to \infty.

Remarks

The transformation rules in point 2 follow from those of the theta functions together with Jacobi's identity θ34=θ24+θ44\theta_3^4 = \theta_2^4 + \theta_4^4, where θ4(τ)=∑n(−1)nqn2\theta_4(\tau) = \sum_{n} (-1)^n q^{n^2}. For example, τ↦τ+1\tau \mapsto \tau + 1 changes θ24\theta_2^4 into −θ24-\theta_2^4 and exchanges θ3\theta_3 and θ4\theta_4, which gives −θ24/θ44=λ/(λ−1)-\theta_2^4/\theta_4^4 = \lambda/(\lambda - 1). The map λ ⁣:ℍ→ℂ∖{0,1}\lambda \colon \mathbb{H} \to \C \setminus \set{0, 1} is a holomorphic covering map. An entire function that omits 00 and 11 lifts through this covering to an entire function with values in ℍ\mathbb{H}. Composing with a Möbius map from ℍ\mathbb{H} onto the unit disc gives a bounded entire function, which is therefore constant; this is the classical proof of Picard's little theorem. The j-invariant is a rational function of λ\lambda: j=256 (1−λ+λ2)3λ2(1−λ)2.j = 256\,\frac{(1 - \lambda + \lambda^2)^3}{\lambda^2 (1 - \lambda)^2}.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectCongruence subgroup, concept, no evidence levelCongruence subgroupconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionTheta function, concept, no evidence levelTheta functionconceptVector space, definition, no evidence levelVector spacedefinitionj-invariant, concept, no evidence levelj-invariantconceptReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSquare matrix, definition, no evidence levelSquare matrixdefinitionSupremum, definition, no evidence levelSupremumdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionDeterminant, definition, no evidence levelDeterminantdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionSpecial linear group, object, no evidence levelSpecial linear groupobjectComplex differentiability, definition, no evidence levelComplex differentiabi…definitionUpper half-plane, object, no evidence levelUpper half-planeobjectHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionEntire function, definition, no evidence levelEntire functiondefinitionPicard's little theorem, statement, no evidence levelPicard's little theor…statementModular lambda function, object, no evidence levelModular lambda functi…object
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Congruence subgroup concept
  3. General linear group object
  4. Identity matrix object
  5. Matrix definition
  6. Möbius transformation concept
  7. Real numbers object
  8. Set bounded above definition
  9. Theta function concept
  10. Vector space definition
  11. j-invariant concept
  12. Real and imaginary parts definition
  13. Square matrix definition
  14. Supremum definition
  15. Linear map definition
  16. Multilinear map definition
  17. Completeness axiom statement
  18. Alternating multilinear map definition
  19. Existence of square roots of nonnegative reals statement
  20. Square root definition
  21. Determinant definition
  22. Modulus of a complex number definition
  23. Special linear group object
  24. Complex differentiability definition
  25. Upper half-plane object
  26. Holomorphic function definition
  27. Entire function definition
  28. Picard's little theorem statement
  29. Modular lambda function object

Further reading

  1. Web https://en.wikipedia.org/wiki/Modular_lambda_function
About this page
Tag4Z1YXT957Z
URIm2://4Z1YXT957Z
Kindobject
WritingW0 stub
MSC11F03, 33E05
UsesUpper half-plane, Theta function, Congruence subgroup, j-invariant, Picard's little theorem
Same as
Changed2 November 2026

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