Lesson plan
Modular lambda function
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 \mathbb{C}/\langle 1, \tau \rangle , where the map is defined as the quotient by the [−1] involution. The q-expansion, where q = e^{\pi i \tau} is the nome, is given by: : \lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots . By symmetrizing the lambda function under the canonical action of the symmetric group S 3 on X(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group \operatorname{SL}_2(\mathbb{Z}) , and it is in fact Klein's modular j-invariant.
Article 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 \mathbb{C}/\langle 1, \tau \rangle , where the map is defined as the quotient by the [−1] involution. The q-expansion, where q = e^{\pi i \tau} is the nome, is given by: : \lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots . By symmetrizing the lambda function under the canonical action of the symmetric group S 3 on X(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group \operatorname{SL}_2(\mathbb{Z}) , and it is in fact Klein's modular j-invariant.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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