geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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.
With the Weierstrass parameterization discussed at elliptic curve – Over general Rings – As solution to the Weierstrass equation, the $j$-invariant is the combination
In the case that 2 and 3 are invertible in the base ring, then this is equivalent to
(…)
Over the complex numbers, with $G_{2k}$ the Eisenstein series and with
and the discriminant
then the $j$-invariant is
(Notice that $1728 = 12^3$.)
(e.g. Miranda 88, def. I.2.1).
Notice the two special values
See at elliptic curve – Definition for general rings – j-invariant.
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.
Over the complex numbers, the $j$-invariant is a map
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
is a modular group($PSL_2(\mathbb{Z})$)-principal bundle and hence classified by a map
from the plane with two points removed or equivalently
(e.g. Miranda 88, section VI.3, p.65)
Wikipedia j-invariant
Rick Miranda, The basic theory of elliptic surfaces, lecture notes 1988 (pdf)
Last revised on December 9, 2015 at 13:29:13. See the history of this page for a list of all contributions to it.