---
created: 2022-02-23T18:45
updated: 2023-03-28T18:02
modification date: Wednesday 23rd February 2022 18:45:08
title: moduli stack of elliptic curves
aliases: [moduli stack of elliptic curves, "M_g", "moduli space of elliptic curves", "moduli space of Riemann surfaces", "Teichmüller space", "moduli space of curves", "Moduli of elliptic curves", moduli of curves]
---

---

- Tags 
	- #AG #higher-algebra/stacks #AG/moduli-spaces 
- Refs:
	- #todo/add-references #todo/too-long 
- Links:
	- [stacks MOC](Unsorted/stacks%20MOC.md)
	- [Unsorted/stacks MOC](Unsorted/stacks%20MOC.md)
	- [moduli space](moduli%20space.md)
	- [representable](representable)
	- [elliptic curve](elliptic%20curve.md)
	- [moduli stack of abelian varieties](moduli%20stack%20of%20abelian%20varieties)
	- [derived moduli stack](derived moduli stack)
	- [Chow ring](Chow%20ring.md)
	- [level structure](level%20structure.md)
	- [cohomological field theory](cohomological%20field%20theory)
	- [Gromov-Witten](Unsorted/Gromov-Witten%20invariants.md)

---

# moduli stack of elliptic curves


# Moduli of curves

- There is an equivalence $\mfh \iso \dcosetr{\SL_2(\RR)}{\SO_2(\RR)}$ as smooth manifolds; this is a moduli space of elliptic curves $E$ with an oriented trivialization of its homology: it represents the functor from complex analytic spaces to sets where $X$ maps to pairs $(f: \mce \to X, \psi: \ZZ^2\iso(\RR^1 f_* \ul\ZZ)\dual)$ where $f$ represents a family of elliptic curves: a smooth proper map, with a section, whose fibers are genus 1 curves.
	- An elliptic curve over an analytic space is equivalent to a polarizable VHS of type $(-1, 0), (0, -1)$.
	- $\mfh$ fits into a theory of [Hermitian symmetric domains](Hermitian%20symmetric%20domains.md), the period domains of Hodge structures.
	- Proving this: look at the stabilizer of $i$.
	- Generalize to $C\dash$valued automorphic forms: $C^0(\dcoset {\Gamma} {\SL_2(\RR)} {\SO_2(\RR)}) \to C)$ (right invariance and left equivariance).
- For $\dcosetr{H}{C}$, one can additionally quotient by $\Gamma \in \Aut(\dcosetr{H}{C})$ to get $\dcoset{\Gamma}{H}{C}$.
	- E.g. $\Gamma(N) \da \ker(\SL_2(\ZZ) \to \SL_2(\ZZ/N\ZZ)) \subseteq \Aut(\mfh)$.
	- See [modular curve](Unsorted/modular%20curve.md)

- $\dcosetr{\SL_n(\CC)}{\SU_n(\RR)}$ is isomorphic to positive-definite Hermitian matrices: $A$ with $A^* = A$, $A>0$, $\det A = 1$.


![](attachments/Pasted%20image%2020220501122226.png)
![](attachments/Pasted%20image%2020220501122340.png)
![](attachments/Pasted%20image%2020220501122423.png)
![](attachments/Pasted%20image%2020220501122530.png)

# Constructing

![2023-03-28-7](2023-03-28-7.png)

# Compatifications

Boundary graphs:
![](attachments/2023-02-08-bgraph.png)
![](attachments/2023-02-08-combinator.png)
![](attachments/2023-02-08-trop.png)

![](attachments/2023-02-08-trop-complex.png)

![](attachments/Pasted%20image%2020220806135220.png)

# Notes

The [moduli space](moduli%20space.md) $\mcm_g$ is not representable in $\Sch$: if it were, then every isotrivial family of curves would be equivalent to a trivial family.
Counterexample to this: take the family
$$
X_t: \quad y^{2}=x^{3}+t, t \neq 0
$$
All fibers are abstractly isomorphic by computing the monodromy action $\pi_1(\CC\units; 1)\actson H^1(X_1, \CC)$

It does admit a [coarse moduli space](coarse%20moduli%20space.md).

**Teichmüller space** is a cover of $M_g$?

## Lattices

![](attachments/Pasted%20image%2020220730133902.png)

## G structures

![](attachments/Pasted%20image%2020220501123136.png)
![](attachments/Pasted%20image%2020220501123234.png)
![](attachments/Pasted%20image%2020220501123334.png)
## Stable Curves
![](attachments/Pasted%20image%2020220208141425.png)
![](attachments/Pasted%20image%2020220208141440.png)
![](attachments/Pasted%20image%2020220208141537.png)
![](attachments/Pasted%20image%2020220208141708.png)
![](attachments/Pasted%20image%2020220208141723.png)

### Stable Reduction

![](attachments/Pasted%20image%2020220208142801.png)

# Homology and Smoothness

![](attachments/Pasted%20image%2020220208150435.png)![](attachments/Pasted%20image%2020220208152343.png)
# Coarse vs fine

![](attachments/Pasted%20image%2020220214101619.png)
![](attachments/Pasted%20image%2020220214101533.png)
![](attachments/Pasted%20image%2020220214101604.png)

# Elliptic curves

![](attachments/Pasted%20image%2020220214102910.png)

# Stable curves and compactification

![](attachments/Pasted%20image%2020220320034324.png)
![](attachments/Pasted%20image%2020220320034333.png)
![](attachments/Pasted%20image%2020220320034342.png)
![](attachments/Pasted%20image%2020220320034854.png)

## Contraction morphism
![](attachments/Pasted%20image%2020220320034457.png)
![](attachments/Pasted%20image%2020220320034521.png)

## Gluing morphism
![](attachments/Pasted%20image%2020220320034532.png)

# Graphs

![](attachments/Pasted%20image%2020220320034649.png)

# Non-representability: nontrivial families

![](attachments/Pasted%20image%2020220730180538.png)

# Deformations and the tangent space

![](attachments/Pasted%20image%2020220806135438.png)


# Operadic structure

View as a [correspondence](Unsorted/correspondences.md):

![](attachments/2023-03-05operad.png)