---
title: modular curve
aliases:
  - modular curves
created: 2022-02-23T18:45
updated: 2024-04-13T21:00
tags:
  - dissertation
---

---

- Tags 
	- #todo/untagged
- Refs:
	- [Notes](attachments/modularCurves2015.pdf)
- Links:
	- [moduli stack of elliptic curves](Unsorted/moduli%20stack%20of%20elliptic%20curves.md)
	- [elliptic curves](MOCs/elliptic%20curve.md)
	- [Fuchsian group](Fuchsian%20group.md)
	- [adele](Unsorted/adelic%20group.md)

---

# modular curve

As a [symmetric space](symmetric%20space.md):

![](2024-04-13-14.png)
![](2024-04-13-15.png)


![](attachments/2023-03-14-1.png)

- $Y_\Gamma(N)$: modular curves with respect to $\Gamma$.
- $X_\Gamma(N)$ their compactifications.
- $\SL_2(\RR)\actson \HH$ by $\matt abcd . z = {az+b\over cz+d}$, 
	- Projectively: $\SL_2(\RR)\actson \PP^1$ by $\matt abcd . \tv{z:1}^t \da [az+b : cz+d]^t = \tv{{a+b\over cz+d}: 1}^t$ in the coordinate chart about zero.
- Elliptic modular curves: all of the form $$Y(N) \da \dcosetl{\Gamma(N)}{\HH}$$
- Importance: coordinates of points on $Y(N)$ generate class fields.
	- Represents a functor $\Sch\to \Set$ which sends $S\mapsto (\mce, \psi_N)$ where $f:\mce \to S$ is an elliptic  curve over $S$ and $\psi_N: \mce[N]\iso C_N^2$ is an isomorphism of $N\dash$torsion. 
	- Yields a smooth connected affine curve as a complex analytic space.
	- Admits a smooth compactification $X(N)$.

- **Theorem**: If $\Gamma\leq \SL_2(\RR)$ is a [Fuchsian group](Unsorted/Fuchsian%20group.md), then $\dcosetl{\Gamma}{\HH}$ is locally compact, connected, Hausdorff, and admits a unique complex structure such that $\HH\to \dcosetl{\Gamma}{\HH}$ is holomorphic.
- **Theorem (Riemann's existence theorem)**: every compact Riemann surface $S$ embeds as $S\embeds \PP^3_\CC$ with image an algebraic curve.
  
## Genus and cusps

![](2023-03-29-1.png)

![](2023-04-06.png)

# Level structures

- Added to a [moduli stack](moduli%20space.md) to remove nontrivial [isotropy](isotropy) groups.
- For curves: given by **congruence subgroups**: used to provide level structure.
	- $$\Gamma(N) \da \ts{M\in \SL_2(\ZZ) \st M\equiv \id \mod N} = \ker \SL_2(\ZZ) \surjects \SL_2(\ZZ/N\ZZ) \leq \SL_2(\ZZ)$$
	- $$\Gamma_1(N) \da ?$$
	- $$\Gamma_0(N) \da ?$$

- For abelian varieties $(A, \lambda)\in \Sch\slice S$:
	- A level $N$ structure is $\phi: A[N]\iso \ul{C_N}\slice S$, an isomorphism of etale sheaves on $S$, plus epsilon.
	- Essentially defined using the Tate module.
- For polarized [K3 surfaces](K3%20surfaces.md)
	- Replace Tate module with $H^2_\et(X_{\kbar}; \ZZladic(1))$
	- Live in open subspaces of the Shimura varieties associated to $\SO_{2, 19}$.
	- Level structures correspond to compact open subgroups $K\leq \SO_{2, 19}(\AA^\fin)$.
	- For primitively polarized K3s of degree 2d, corresponding to subgroups of $\SO(V_{2d}, \psi_{2d})(\AA^\fin)$ where $(V_{2d}, \psi_{2d})$ is the quadratic [[lattice]] associated to $L_{2d} \da U\sumpower 2 \oplus E_8(-1)\sumpower 3 \oplus \gens{e_1 + df_1}$ where $\ts{e_i, f_i}$ is a symplectic basis of $U$.
- Drinfeld level structures: see Katz-Mazur.

![](attachments/Pasted%20image%2020220807155534.png)

![](attachments/2023-02-27abvarlevel.png)

# As moduli spaces


- As groupoids, where morphisms are $f: E\to E'$ satisfying conditions.
	- $Y(N):$ pairs $(E, \phi_N)$ where $E$ is elliptic and $\phi_N: C_N^2 \to E[n]\in \Grp$.
		- Morphisms: $f\circ \phi_E = \phi_{E'}$
	- $Y_1(N):$ pairs $(E, p)$ where $E$ is elliptic and $p\in E(k)$ is a point of order $N$.
		- Morphisms: $f(p_E) = p_{E'}$.
	- $Y_0(N):$ pairs $(E, H)$ where $E$ is elliptic and $H\leq E(k)$ is a cyclic subgroup of order $N$.
		- Morphisms: $f(H_E) = H_{E'}$.

- For $N\geq 4$, $X_0(N)$ is a coarse moduli space and $X_1(N)$ is a fine moduli space (over $\QQ$)

# Modular forms


- Cups forms of weight 2 for $\Gamma(N) \mapstofrom$ holomorphic differential forms on $X(N)$.
- Spaces $\mcm_n$ of modular forms admit actions by the ring of Hecke operators, so eigenforms make sense.
	- For $f(z) \da \sum a_n q^n$ a new cuspidal eigenform, each $a_n$ generates some $K\in \NF$.
	- $L(f, s) \da \sum a_n n^{-s}$ and $L(\rho_f, s) = \prod_p E_p$ where each $E_p$ is an Euler factor, almost all of which are of the form $$E_p = \det \qty{1-\rho_f(\Frob_p) p^{-s} }\inv$$
- **Theorem (Eichler-Shimura, Deligne)**: there is an associated family of Galois representations $\rho_f: G_\QQ\to \bar\QQladic$ for each embedding $E\embeds \bar\QQladic$ yielding an equality of [L functions](Unsorted/L%20function.md) $$L(f, s) = L(\rho_f, s)$$
	- Interpret this theorem as *non-abelian reciprocity*.
	- Proved by studying $H^1_\et(X(N)\slice{\QQbar}; \bar\QQladic) \in \Rep(G_\QQ)^\fd$.