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created: 2022-04-05T23:42
updated: 2023-12-28T18:14
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---
date: 2021-10-21 18:42
modification date: Thursday 21st October 2021 20:41:02
title: modular form
aliases: ["modular forms", "modularity", "modular"]

---

Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #arithmetic-geometry/Langlands #NT/analytic 
- Refs:
	- Course notes: <https://www.math.columbia.edu/~phlee/CourseNotes/ModularForms.pdf#page=1> #resources/course-notes 
	- Apostol, *Modular Functions and Dirichlet Series in Number Theory*.
	- See also previous book in series, *Introduction to Analytic Number Theory*
	- Harvard summer tutorial: <https://people.math.harvard.edu/~smarks/mod-forms-tutorial/index.html>
- Links:
	- [Hecke operator](Hecke operator)
	- [Eisenstein series](Eisenstein series)
	- [Weight of a modular form](Weight of a modular form)
	- [modular curve](Unsorted/modular%20curve.md)
	- [Siegel modular forms](Siegel%20modular%20forms)
	- [Hecke operator](Hecke%20operator)
	- [automorphic form](Unsorted/automorphic%20form.md)

---

# modular form

![](attachments/2023-01-12modform.png)
![](attachments/2023-01-12cusp.png)
![](attachments/2023-01-12universalmodular.png)


![](attachments/2023-01-12modularform.png)


![](attachments/Pasted%20image%2020220430213647.png)

- Ways to think of a modular form:
	- Functions ([automorphic forms](automorphic%20forms)) on $\GL_2$
	- Functions on $\HH$
	- Sections of a line bundle over a moduli of curves

- Related to [q series](q%20series.md)
	- Many classical modular forms are [generating function](generating function) for integer partitions in interesting ways

![](attachments/Pasted%20image%2020220211002118.png)
![](attachments/Pasted%20image%2020220210182831.png)
![](attachments/Pasted%20image%2020220211002301.png)

# L functions

![](attachments/Pasted%20image%2020220430213719.png)

# Examples

- Ramanujan delta: $\Delta(q)=q \prod_{n \geq 1}\left(1-q^{n}\right)^{24}$, a holomorphic cusp form of [weight](weight%20of%20a%20modular%20form) 12 and [level](level%20of%20a%20modular%20form) 1.



# Notes

- **Modularity theorem**:
	If $E \in \Ell_{\QQ}$, then $E$ admits a rational parameterization. Proved by Wiles et al.

![](attachments/Pasted%20image%2020220202221424.png)
	![Pasted image 20211029131059.png](Pasted%20image%2020211029131059.png)
	![Pasted image 20211029131114.png](Pasted%20image%2020211029131114.png)
	![Pasted image 20211029131212.png](Pasted%20image%2020211029131212.png)

![](attachments/Pasted%20image%2020220204094501.png)

Relation to [Weil Conjectures](Unsorted/Weil%20Conjectures.md):
![](attachments/Pasted%20image%2020220204094635.png)

# Stack formulation

![](attachments/2023-01-12stack.png)

# Modular forms from lattices

Let $L$ be a lattice of signature $(2, n)$ for $n\geq 1$, let $\Omega_L$ be the associated period domain, let $\Orth^+(L) \leq \Orth(L)$ be the subgroup preserving $\Omega_L$ and let $\tilde \Omega_L$ be the affine cone over $\Omega_L$.
Let $n\geq 3$, let $k\in \ZZ$, and let $\Gamma \leq \Orth^+(L)$ be a finite-index subgroup with $\chi: \Gamma\to \CC^*$ a character. A holomorphic functional $f: \Omega_L \to \CC$ is called a **modular form of weight $k$ and character $\chi$ for $\Gamma$** if

1.  Factor of automorphy: $f(\lambda z) = \lambda^{-k} f(z)$ for any $\lambda \in \CC^*$.

2.  Equivariance: $f(\gamma z) = \chi(\gamma) f(z)$ for all $\gamma\in \Gamma$.