---
date: 2021-04-18
tags: [ web/quick-notes ]
---


# 2021-04-18

## Notes on modular forms

#arithmetic-geometry/modular-forms 

- Modular forms can be defined as functions $\HH\to \CC$ satisfying weak $\Gamma\dash$invariance.
- Also sections of a bundle: the [modular curve](modular%20curve.md).
- [weight of a modular form](weight%20of%20a%20modular%20form) : refers to growth rates of these functions.
  - A weight $k$ modular form is an element of $H^0(X; \omega^{\tensor k})$ where $X$ is the compactified modular curve, a quotient of $H \union \PP^1(\QQ)$
    - This definition extends to $H/\Gamma$

- Weird fact: $M_1$ is one-dimensional, but for $g\geq 2$ we have $\dim M_g = 3g-3$

  - Special things for $g=1$: $q \dash$expansions (i.e. Fourier series), vanishing [Torelli](Torelli.md), $\pi_1 \TT$ for the torus is abelian, the $\theta$ function has a discrete zero locus, infinite product expansions like Jacobi's triple product

- Higher genus generalizations come not from a Teichmüller cover $T_g \surjects M_g$ or $M_g$, no one seems to care about those though.
  - People do care about [Siegel modular forms](Siegel%20modular%20forms.md) : replace $\HH$ with $\HH_S^g$ the symmetric $g\times g$ matrices with positive-definite imaginary part

- $\HH_S^g/\Sp(2g; \ZZ)$ is somehow a model for [moduli stack of abelian varieties](moduli%20stack%20of%20abelian%20varieties), $M_g$ embeds as a variety since we have the [Jacobian](Jacobian.md)

- [Hodge bundle](Hodge%20bundle.md) : rank $g$ over $M_g$, fibers over isomorphism classes are $H^0(X, K_X)$ where $K$ is the [canonical bundle](canonical%20bundle.md), then take the determinant bundle.

  - Surprisingly, $\HH_S^g$ is a [Lie group](Lie%20group.md) but not a [Lie algebra](Lie%20algebra.md) : $[AB]^t = -[BA]^t$, so it's not closed.