group cohomology, nonabelian group cohomology, Lie group cohomology
principal ?-bundle?
covering ?-bundle?/local system
cohomology with constant coefficients / with a local system of coefficients
?-Lie algebra cohomology?
differential cohomology
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ?-stacks?
fundamental ?-groupoid in a locally ?-connected (∞,1)-topos? / of a locally ?-connected (∞,1)-topos?
derived smooth geometry
This is a sub-entry of
and
see there for background and context.
This entry contains a basic introduction to elliptic curves and their moduli spaces.
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A Survey of Elliptic Cohomology - formal groups and cohomology
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the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention, meaning: somebody should go through this and polish
Definition An elliptic curve over $\mathbb{C}$ is equivalently
a Riemann surface $X$ of genus 1 with a fixed point $P \in X$
a quotient $\mathbb{C}/\Lambda$ where $\Lambda$ is a lattice in $\mathbb{C}$;
a compact complex Lie group of dimension 1.
a smooth algebraic curve of degree 3 in $\mathcal{P}$.
Remark The third definition is the one that is easiest to generalize. For our simple purposes, though, the second one will be the most convenient.
From the second definition it follows that to study the moduli space of elliptic curves it suffices to study the moduli space of lattices in $\mathbb{C}$.
Definition A framed elliptic curve is an elliptic curve $(X,P)$ (in the sense of the first definition above) together with an ordered basis $(a,b)$ of $H_1(X, \mathbb{Z})$ with $(a \cdot b) = 1$
A framed lattice in $\mathbb{C}$ is a lattice $\Lambda$ together with an ordered basis $(\lambda_1, \lambda_2)$ of $\Lambda$ such that $Im(\lambda_2/\lambda_1) \gt 0$.
(So turning $\lambda_1$ to $\lambda_2$ in the plane means going counterclockwise).
this implies that
the upper half plane $\mathfrak{h}$ is in bijection with framed lattices in $\mathbb{C}$ which in turn is in bijection with isomorphism classes of framed elliptic curves over $\mathbb{C}$
and we have
where $SL_2(\mathbb{Z}) = \left\{ \left(\array{a & b \\ c & d }\right)| a d - c d = 1\right\}$ acts by
Claim the quotient $\mathfrak{h}/_{SL_2(\mathbb{Z})}$ is biholomorphic to the disk and has a unique structure of a Riemann surface which makes the quotient map $\mathfrak{h} \to \mathfrak{h}/SL_2(\mathbb{Z})$ a holomorphic map
warning possibly something wrong here, audience doesn’t believe the bit about the disk
definition write $M_{1,1} := \mathfrak{h}/SL_2(\mathbb{Z})$
definition a homolorphic family of elliptic curves over a complex manifold $T$ is
a holomorphic map $\pi : X \to T$
together with a section $s : T \to X$ of $\pi$ such that for any $t \in T$ the pair $(X_t, s(t))$ is an elliptic curve (using the first definition above).
For every family
we would like to have $F \to M_{1,1}$
where
such that
$\phi : T \to M_{1,1}$ is a holomorphic map
every holomorphic map $T \to M_{1,1}$ corresponds to a family over $t$;
there is a universal family over $M_{1,1}$
This is impossible . One can construct explicit counterexamples. These counterexamples involved elliptic curves with nontrivial automorphisms.
For instance
but see the discussion at moduli space for a discussion of the statement “it’s te automorphisms that prevent the moduli space from existing”
consider
given by
Then consider the family
is a family of elliptic curves over $\mathfrak{h}$
and $E_\tau = \mathbb{C}/{\Lambda_\tau}$ with
is a family of framed elliptic curves.
fact the space $\mathfrak{h}$ with the family $E \to \mathfrak{h}$ is a fine moduli space for framed elliptic curves.
Consider any map $\phi : T \to \mathfrak{h}$
with pullback of the universal family
claim for every point $t \in T$ there is an open neighbourhood $t_0 \in U \hookrightarrow T$ such that one can choose 1-forms $\omega_t$ on $X_\tau$ which vary holomorphically with respect to $t$.
Notice that locally every family of elliptic curves is framed (since we can locally extend a choice of basis for $H_1$). So
at $i$ and $\rho = e^{2\pi i/6}$ , $C = \{\pm I\}$
isn’t locally liftable at $i$ and $\rho$ so it is not a univresal family of unframed curves.
definition A basic pointed orbifold (basic meaning global) is a triple $X//\Gamma := (X,\Gamma,\rho)$, where
$X$ is a connected and simply connected topological space (or in other variants a complex manifold or whatever is under consideration)
$\Gamma$ is a discrete group
$\rho : \Gamma \to Aut(X)$ is a group homomorphism
(here “pointed” because we specified the action $\rho$ instead of its iso-class under the following morphisms)
A morphism from $(X,\gamma, \rho)$ to $(X', \Gamma', \rho')$ is a pair
where
$f : X \to X'$ is a continuous map
$\phi : \Gamma \to \Gamma'$ is a group homomorphism
such that for all $\gamma \in \Gamma$
This really leads an enlargement of the plain category of spaces:
remark We have a faithful embedding of spaxces into orbifolds defined this way: for any connected semi-locally simply connected space $X$ with universal cover $\tilde X$ we have
warning notice all the simply-connectedness assumoptions above for making sense of this
remark let $X$ be a nice topological space. Let $G = \pi_1(X)$ be its first homotopy group and let a discrete group $\Gamma$ action on $X$.
then define
then we have an exact sequence
where $G \to \tilde \Gamma$ is given by $(g \mapsto (1,g))$ and $\tilde \Gamma \to \Gamma$ by $(\gamma,g) \mapsto \gamma$.
definition
For an orbifold $(X,\Gamma,\rho)$ write $I \times (X,\Gamma,\rho) := (I \times X, \Gamma, \rho)$.
Then a homotopy from $(f,\phi)$ to $(f',\phi') : (X,\Gamma, \rho) \to (X',\Gamma', \rho')$
is a map
such that
$\Psi = \phi = \phi'$
$f(-) = F(0,-)$, $f'(-) = F(1,-)$
now write
(the circle regarded as a global orbifold)
definition
The first homotopy group for our definition of orbifold is:
exercise show that this is $\cdots \simeq \Gamma$
(recall again the simply-connectness assumoption!!)
definition* A morphism
is a weak homotopy equivalence if $\phi$ is an ismorphism and $H_\bullet(f) : H_\bullet(X) \to X_\bullet(X')$.
note Let $E \Gamma$ be a contractible space on which $\Gamma$ acts properly, dic. and free, then
with $\phi = Id_\Gamma$ and $f$ the projection is a weak homotopy equivalence.
definition a local system $V$ on $(X,\Gamma)$ with fiber $V$ a group homomorphism $\Gamma \to Aut(V)$ with
definition
Introduce the following notation for homotopy groups, homology and integral cohomology of our orbifolds with coefficients in a local system:
$\pi_n(X//\Gamma)$ := \pi_n(X)$for$n \geq 2$
$H_\bullet(X//\Gamma, V) := H_\bullet(E \Gamma \times_\Gama X, V)$
$H^\bullet(X//\Gamma, V) := H^\bullet(E \Gamma \times_\Gama X, V)$
example ${*}//\Gamma$ has a weak homotopy equivalence to the classifying space $\mathcal{B}\Gamma$
it follows that for local system $V$ we have
where on the right we have group cohomology
We have all kinds of constructions on orbifolds by saying they are structures on $X$ with suitable extension of the action of $\Gamma$ to them
A vector bundle on an orbifold $V \to X//\Gamma$ is a vector bundle $V \to X$ with isomorphism action by $\Gamma$ specified, covering that on $X$.
for instance the tangent bundle of $X //\Gamma$ is given by $(T X)//\Gamma \to X//\Gamma$ in the obvious way.
definition say that $\Gamma$ acts virtually freely if $\exists$ a finite index subgroup $\Gamma'$ of $\Gamma$ which acts freely on $X$.
note $SL_2(\mathbb{Z})$ acts virtually freely on $\mathfrak{h}$
Let $\Gamma' \lt \Gamma$ be a finite index subgroup which acts freely on $X$.
set
the map
must be viewed as an unramified covering of degree $[\Gamma:\Gamma']$.
supposedly important statement
definition
the Euler characteristic of a global orbifold is
compare groupoid cardinality
definition
Define now the global orbifold
proposition
and similarly for integral cohomology
Since the upper half plane is contractible, the homotopy type of $\mathfrak{h}//\mathbb{Z}_2$ are the same as that of $* // \mathbb{Z}_2$ and similarly for the (group)cohomology
and similalry for homology.
In particular
for all $m \in \mathbb{N}$ then
so that
fact the group $SL_2(\mathbb{Z})[m]$ is free for $m \gt 2$.
so far all $\mathbb{Q}$-representations $V$ we have
due to the freeness we have also that
for $k \geq 2$
and hence
is torsion
proposition
as orbifolds, we have an isomorphism
where
and $S_3$ acts on that by permuting $0,1, \infty$. (Think of $\mathbb{P}^1$ as the Riemann sphere: there is a unique holomorphic automorphism of that permuting these three points in a given fashion.) While $C_2$ acts trivially.
proof*
now $PSL_2(\mathbb{Z})[2]$ is known to be torsion free. It acts in a standard way on the upper half plane $\mathfrak{h}$.
A little discussion shows that
this implies that
the free group on two generators.
Then the second but last map
has a section, from which we get that
and so
which is the end of the proof.
corollary The Euler characteristic of the moduli stack of elliptic curves is
now consider the line bundle
with action on the total space for $k \in\mathbb{Z}$
call this line bundle on the moduli stack $\mathcal{L}_k \to \mathcal{M}_{1,1}$. We will see that all line bundles are isomorphic to one of these.
remark
is a section of $\mathcal{L}_k$ iff
hence precisely if it defines a modular function of weight $k$! This gives a geometric interpretation of modular functions.
and define an action of $G := SL_2(\mathbb{Z}) \letimes \mathbb{Z}^2$
where $\mathbb{Z}^2$ acts on $SL_2(\mathbb{Z})$ by
and on $\mathbb{C} \times \mathfrak{h}$ by
the resulting bundle
we call
theorem for any complex manifold $T$ there is a bijection between families of elliptic curves over $T$ and orbifold maps $T \to \mathcal{M}_{1,1}$ classify them.
Suppose we have an “isotrivial family” (meaning all fibers are isomorphic elliptic curves, i.e. a fiber bundle of elliptic curves)
recall that the group that defines $T$ as an orbifold is the first homotopy group $\pi_1(T)$.
The only condition that we get from the definition of orbifold maps is that
factors through the stabilizer group $\simeq Aut(E_p)$ of our base point $p \in \mathcal{M}_{1,1}$
one can see that over compact $T$ with $\mathcal{M}_{1,1}$ we cannot have nontrivial famlies without singular fibers.
To get around that we want a compactification $\bar \mathcal{M}_{1,1}$ of the moduli stack.
also fur purposes of intersection theory, we need to further compactify.
recall the description of $\mathcal{M}_{1,1}$ as a weak quotient of $\mathbb{P}^1$. Then consider:
definition
Let
otice that this is now an orbifold which is no longer basic by the above definition. In fact, we can cover it by charts of basic orbifolds as follows: consider
with the arrows being maps of orbifolds whose precise details I haven’t typed.
then let $\mathbb{D}^*$ be the punctured disk and realize the diagram
where the right morphism is just the inclusion
now we build a chart of $\bar \mathcal{M}_{1,1}$ consisting of the two patches $\mathcal{M}_{1,1}$ and $\mathbb{D}/C_2/$
from this we get the alternative
definition
the colimit on the right manifestly glues in the “point at infinity” that is not hit by the map $\mathbb{D}^*//C_2 \to \mathcal{M}_{1,1}$.
definition A stable curve (over $\mathcal{C}$) of genus $g$ with $n$ marked points is a proper, connected curve with $n$ smooth marked points such that all singularities are nodes and such that the the automorphism group (of autos respecting the smooth marked points) is finite,
and such that the arithmetic genus is $g$.
Now $\bat \mathcal{M}_{g,n}$ is the fine moduli space for smooth curves of genus $g$.
There is a line bundle
built fiberwise from the cotangent spaces of the elliptic curves.
one of them is obtained from one of the $n$ sections $s_i$ of the universal family $\mathcal{F} \to \bar \mathcal{M}_{g,n}$. The fiber over a point is the cotangent space of the elliptic curve over that point at this section.
Write for the first Chern class
such that
then we get numbers called the Gromov-Witten invariants (“of the point”)
Let $x, y$ by affine coordinates on $\mathbb{P}^2$
Let $f(x,y)$ and $g(x,y)$ be two generic cubics, in particular there are nine joint zeros
called $p_1, \cdots, p_9$.
define then
and consider
That map $q$ has degree $\frac{1}{2}$ (!) since $\mathbb{P}^1 \to \bar \mathcal{M}_{1,1}$ has degree 12
we also find that the diaginal map $\mathbb{P}^1 \to \bar M_{1,1}$ has degree 12. It follows that $\phi$ has degree 24:
Now let $\mathcal{T}_i^* \to \bar \mathcal{M}_{g,n}$ be one of these line bundles. Consider the pullback $\phi^*(\mathcal{T}_1)$
then by some argument not reproduced here we find
Then since the order of $\phi$ is 24 we find that the first Gromov-Witten invariant is
recall that the moduli stack of elliptic curve is, as a global orbifold
and also
and there is a line bundle on this given by
where the action is given by
since $SL_2(\mathbb{Z})^{ab} = \mathbb{Z}/12\mathbb{Z}$ one finds the Picard group
meromorphic (holomorphic) sections $f$ of $\mathcal{L}_k$ are modular functions of weight $k$, i.e. $f : \mathfrak{h} \to \mathbb{C}$ such that
the universal elliptic curve over $\mathcal{M}_{1,1}$ is
Then we ended last time with describing the compactified moduli space
proposition $\mathbb{L}_k$ has a universal extension $\bar \mathcal{L}_k$ to $\bar \mathcal{M}_{1,1}$
proof
take
where $C_2$ acts by
note that since $\left(\array{1 & n \\ 0 1}\right) \in S_2(\mathbb{Z})$ for all $n \in \mathbb{Z}$ ay modular function $f : \mathfrak{h} \to \mathbb{C}$
where $q := e^{2 \pi i \tau}$
is called a holomorphic modular form of weight $k$ if $f : \mathfrak{h} \to \mathbb{C}$ is holomorphic and $a_n = 0$ for all $n \lt 0$
remark modular forms of weight $k$ are in bijection with sections of the line bundle $\bar \mathcal{L}_k$.
example for any lattice $\Lambda$ in $\mathbb{C}$ and for any $k \gt 2$ we have
obviously for all $u \in \mathbb{C}^*$ $S_k(u \Lambda) = u^{-k} S_k(\Lambda)$
and for all $\tau \in \mathfrak{h}$ with $C_k(\tau) := S_k(\Lambda_\tau)$ it follows that $G_k : \mathfrak{h} \to \mathbb{C}$ is holomorphic
since $\Lambda_{\gamma \tau} = (c \tau + d)^{-1} \Lambda_\tau$ it follows that $G_k$ is a modular function of weight $k$
fact $G_{2k} = 2 \zeta(2 k) ü 2 \frac{(2 \pi i)^{2k}}{(2k-1)!} \sum_{n=1}^\infty b_{2k-1}(n)q^n$
with $b_k(n) := \sum_{d|n} d^k$ and where $\zeta$ is the zeta-function?
it follows that
$G_{2k}$ is a modular form of weight $2k$ (which is not a cusp form).
an important cusp form is
setting $g_2 = 60 G_4$ and $g_3 = 140 G_6$
the modular form
is a cusp form of weight 12. $\Delta$ does not have any 0 in $\mathfrak{h}$ and it has a simple zero at $q = 0$.
we have an isomorphism
where on the right is the sheaf with at most a pole at $\infty$. This isomorphism going from right to left is induced by multiplication with $\Delta$.
we have an exact sequence
where the first nontrivial map sends 1 to $\bar \mathcal{L}_{12}$ and the second one $\bar \mathcal{L}_1$ to the generator.
set for all $k$
proposition $M_\bullet$ is an even graded algebra freely generated by $G_4$ and $G_6$ and the ideal $M_\bullet^\circ$ is generated by $\Delta$.
the dimensions are
dim M_{2k} =
\left\{
floor k/6 & for k = 1 mod 6
\\
1+ floor k/6 & otherwise
\right.
???????
Recall the three definition of elliptic curves from above.
Now a fourth definition:
definition an elliptic curve is a smooth curve of degree 3 in $dim M_{2k} = \left\{ floor k/6 &\mathbb{C}\mathbb{P}^2$ together with a point in it.
that this equation implies the first one above follows from the genus formula, which says that a degree $n$ curve as in the definition has genus $g = \frac{(n-1)(n-2)}{2}$
that the first def implies this one