j-invariant in nLab

Contents

Contents

Idea

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

Over the complex numbers the jj-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 jj-invariant is the combination

j≔c 4 3Δ. 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 2kG_{2k} the Eisenstein series and with

g 2≔60G 4 g_2 \coloneqq 60 G_4

g 3≔140G 6 g_3 \coloneqq 140 G_6

and the discriminant

Δ≔g 2 3−27g 3 2 \Delta \coloneqq g_2^3 - 27 g_3^2

then the jj-invariant is

j=1728g 2 3Δ. j = 1728 \frac{g_2^3}{\Delta} \,.

(Notice that 1728=12 31728 = 12^3.)

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

Notice the two special values

(j=0)⇔(g 2=0) (j = 0 ) \Leftrightarrow (g_2 = 0)

(j=1728)⇔(g 3=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=0j = 0, hence g 2=0g_2 = 0, which is the one given by quotienting out an equilateral lattice; this has automorphism group ℤ/6ℤ\mathbb{Z}/6\mathbb{Z};

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

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

The case j→∞j \to \infty, hence Δ=0\Delta = 0 but g 2≠0g_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 jj-invariant is a map

j:𝔥⟶ℂ 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∈ℂ0,\;1728 \in \mathbb{C}.

The induced unramified covering

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

is a modular group(PSL 2(ℤ)PSL_2(\mathbb{Z}))-principal bundle and hence classified by a map

ℂ−{0,1728}⟶BPSL 2(ℂ) \mathbb{C} - \{0,1728\} \longrightarrow B PSL_2(\mathbb{C})

from the plane with two points removed or equivalently

π 1(ℂ−{0,1728})⟶PSL 2(ℤ). \pi_1(\mathbb{C}-\{0,1728\}) \longrightarrow PSL_2(\mathbb{Z}) \,.

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

• 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)