complex geometry

# Contents

## 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

$(j = 0 ) \Leftrightarrow (g_2 = 0)$
$(j= 1728) \Leftrightarrow (g_3 = 0)$

## 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

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