nLab j-invariant

Idea

What is called the $j$-invariant is an invariant of cubic curves and hence of elliptic curves, partly characterizing them.

Over the complex numbers the $j$-invariant is a modular function on the upper half plane which serves to characterize most of the properties of the moduli stack of elliptic curves in this case.

Definition

Over a general ring

With the Weierstrass parameterization discussed at elliptic curve – Over general Rings – As solution to the Weierstrass equation, the $j$-invariant is the combination

$$j \coloneqq \frac{c_4^3}{\Delta} \,.$$

In the case that 2 and 3 are invertible in the base ring, then this is equivalent to

(…)

Over the complex numbers

Over the complex numbers, with $G_{2k}$ the Eisenstein series and with

$$g_2 \coloneqq 60 G_4$$

$$g_3 \coloneqq 140 G_6$$

and the discriminant

$$\Delta \coloneqq g_2^3 - 27 g_3^2$$

then the $j$-invariant is

$$j = 1728 \frac{g_2^3}{\Delta} \,.$$

(Notice that $1728 = 12^3$.)

(e.g. Miranda 88, def. I.2.1).

Notice the two special values

$$(j = 0 ) \Leftrightarrow (g_2 = 0)$$

$$(j= 1728) \Leftrightarrow (g_3 = 0)$$

General

See at elliptic curve – Definition for general rings – j-invariant.

Properties

Characterization of complex elliptic curves

Over the complex numbers, there are two elliptic curves with special values

• the curve with $j = 0$, hence $g_2 = 0$, which is the one given by quotienting out an equilateral lattice; this has automorphism group $\mathbb{Z}/6\mathbb{Z}$;

• the curve with $j = 1728$, which corresponds to dividing out the lattice $(1,i)\mathbb{Z}$, this has automorphism group $\mathbb{Z}/4\mathbb{Z}$.

All other curves have automorphism $\mathbb{Z}/2\mathbb{Z}$, given by inversion involution.

The case $j \to \infty$, hence $\Delta = 0$ but $g_2 \neq 0$, corresponds to the nodal curve which is added in the Deligne-Mumford compactification of the moduli stack of elliptic curves.

As a branched cover of the complex plane

Over the complex numbers, the $j$-invariant is a map

$$j \;\colon \; \mathfrak{h} \longrightarrow \mathbb{C}$$

from the upper half plane to the complex numbers. This is a branched cover, with two branching points being $0,\;1728 \in \mathbb{C}$.

The induced unramified covering

$$j \;\colon\; (\mathfrak{h}-j^{-1}(\{0,1728\})) \longrightarrow (\mathbb{C}-\{0,1728\})$$

is a modular group($PSL_2(\mathbb{Z})$)-principal bundle and hence classified by a map

$$\mathbb{C} - \{0,1728\} \longrightarrow B PSL_2(\mathbb{C})$$

from the plane with two points removed or equivalently

$$\pi_1(\mathbb{C}-\{0,1728\}) \longrightarrow PSL_2(\mathbb{Z}) \,.$$

(e.g. Miranda 88, section VI.3, p.65)

Related concepts

• moduli space of elliptic curves

• elliptic genus

• Moonshine, Monster vertex algebra

References

• Wikipedia j-invariant

• Rick Miranda, The basic theory of elliptic surfaces, lecture notes 1988 (pdf)