---
tag: 4Z1YXT957Z
kind: object
name: Modular lambda function
msc: [11F03, "33E05"]
uses: [X6XGGH5FPW, 2F5CYQJVAF, 1H6R856P3C, FEC2XH822R, R2PHM118NG]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_lambda_function
same_as:
  wikidata: Q6889722
  wikipedia: Modular lambda function
---

## Description

Let $\mathbb{H} = \set{\tau \in \C : \operatorname{Im}\tau > 0}$ be the [[upper half-plane|X6XGGH5FPW]], and for $\tau \in \mathbb{H}$ let $q = e^{\pi i \tau}$.
Let $\theta_2$ and $\theta_3$ be the [[theta functions|2F5CYQJVAF]]
$$\theta_2(\tau) = \sum_{n \in \Z} q^{(n + 1/2)^2}, \qquad \theta_3(\tau) = \sum_{n \in \Z} q^{n^2}.$$
The *modular lambda function* is
$$\lambda(\tau) = \frac{\theta_2(\tau)^4}{\theta_3(\tau)^4}, \qquad \tau \in \mathbb{H}.$$

1. $\lambda$ is holomorphic on $\mathbb{H}$ and never takes the values $0$ and $1$.
2. $\lambda(\tau + 1) = \dfrac{\lambda(\tau)}{\lambda(\tau) - 1}$ and $\lambda(-1/\tau) = 1 - \lambda(\tau)$.
3. $\lambda$ is invariant under the principal [[congruence subgroup|1H6R856P3C]] $\Gamma(2)$ of level $2$.
4. $\lambda(\tau) = 16q + O(q^2)$ as $\operatorname{Im}\tau \to \infty$.

## Remarks

The transformation rules in point 2 follow from those of the theta functions together with Jacobi's identity $\theta_3^4 = \theta_2^4 + \theta_4^4$, where $\theta_4(\tau) = \sum_{n} (-1)^n q^{n^2}$.
For example, $\tau \mapsto \tau + 1$ changes $\theta_2^4$ into $-\theta_2^4$ and exchanges $\theta_3$ and $\theta_4$, which gives $-\theta_2^4/\theta_4^4 = \lambda/(\lambda - 1)$.
The map $\lambda \colon \mathbb{H} \to \C \setminus \set{0, 1}$ is a holomorphic covering map.
An entire function that omits $0$ and $1$ lifts through this covering to an entire function with values in $\mathbb{H}$.
Composing with a Möbius map from $\mathbb{H}$ onto the unit disc gives a bounded entire function, which is therefore constant; this is the classical proof of [[Picard's little theorem|R2PHM118NG]].
The [[j-invariant|FEC2XH822R]] is a rational function of $\lambda$:
$$j = 256\,\frac{(1 - \lambda + \lambda^2)^3}{\lambda^2 (1 - \lambda)^2}.$$

## Explanations

### L1 human

The lambda function is a holomorphic function on the upper half-plane with many symmetries.
It never takes the values $0$ and $1$, and every other complex value is taken infinitely often.
This makes it the tool that shows a nonconstant entire function can miss at most one complex value.