The moduli stack of elliptic curves.
For the moment see A Survey of Elliptic Cohomology - elliptic curves for more.
An elliptic curve is determined, up to non-canonical isomorphism, by its j-invariant
Here every complex number appears as a value, and therefore the moduli space of elliptic cuves a priori is not compact.
A compactification of the moduli space is obtained by including also elliptic curves with nodal singularity?.
Description over the complex numbers
Upper half plane
The upper half plane is in bijection with framed lattices in the complex plane , which in turn is in bijection with isomorphism classes of framed elliptic curves over
and we have
where the special linear group over the integers
The naive moduli space and its problems
for the plain quotient of the upper half plane by the above group action.
For every family
we would like to have such that there is a pullback
is a holomorphic map
every holomorphic map corresponds to a family over ;
there is a universal family over
This is impossible . One can construct explicit counterexamples. These counterexamples involve elliptic curves with nontrivial automorphisms.
but see the discussion at moduli space for a discussion of the statement “it’s te automorphisms that prevent the moduli space from existing”
Moduli space of framed elliptic curves
Then consider the family
is a family of elliptic curves over
is a family of framed elliptic curves.
The space with the family is a fine moduli space for framed elliptic curves?.
Consider any map
with pullback of the universal family
claim for every point there is an open neighbourhood such that one can choose 1-forms on which vary holomorphically with respect to .
Notice that locally every family of elliptic curves is framed (since we can locally extend a choice of basis for ). So
at and ,
isn’t locally liftable at and so it is not a univresal family of unframed curves.
Moduli stack/orbifold of elliptic curves
This is the moduli stack of elliptic curves.
Compactified moduli stack
Consider the complex analytic parameteritation over the annulus
of elliptic curves
This has an extension to the origin, where is a nodal curve. Algebraically, in a formal neighbourhood? of the origin, hence over , this is the Tate curve.
e.g. (Lurie, section 4.3).
and similarly for integral cohomology
An introduction is for instance in