# Contents

## Idea

A modular form is a holomorphic function on the upper half-plane that satisfies certain transformation properties.

Modular forms appear as

## Definition

### In components

###### Definition

An (integral) modular form of weight $w$ is a holomorphic function on the upper half-plane

$f:\left({ℝ}^{2}{\right)}_{+}↪ℂ$f : (\mathbb{R}^2)_+ \hookrightarrow \mathbb{C}

(complex numbers with strictly positive imaginary part)

such that

1. if $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_{2}\left(ℤ\right)$ acting by $A:\tau ↦=\frac{a\tau +b}{c\tau +d}$ we have

$f\left(A\left(\tau \right)\right)=\left(c\tau +d{\right)}^{w}f\left(\tau \right)$f(A(\tau)) = (c \tau + d)^w f(\tau)

(notice that for $A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ then $f\left(\tau +1\right)=f\left(\tau \right)$)

2. $f$ has at worst a pole at $\left\{0\right\}$ (for weak modular forms this condition is relaxed)

it follows that $f=f\left(q\right)$ with $q={e}^{2\pi i\tau }$ is a meromorphic funtion on the open disk.

3. integrality $\stackrel{˜}{f}\left(q\right)={\sum }_{k=-N}^{\infty }{a}_{k}\cdot {q}^{k}$ then ${a}_{k}\in ℤ$

### Abstractly

More abstractly, for ${ℳ}_{\mathrm{ell}}$ the moduli stack of elliptic curves and $A\to {ℳ}_{\mathrm{ell}}$ the corresponding universal bundle, write ${\Omega }_{A/S}^{1}$ for the line bundle of fiberwise Kähler differential forms. Write $e$ for the 0-section of this line bundle. Then

$\omega ≔{e}^{*}{\Omega }_{A/S}^{1}$\omega \coloneqq e^\ast \Omega^1_{A/S}

is a line bundle over the moduli stack of elliptic curves. A modular form of weight $k$ is a section of ${\omega }^{\otimes k}$

## Examples

• Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, Don Zagier, The 1-2-3 of modular forms, Lectures at a Summer School 2004 in Nordfjordeid, Norway; Universitext, Springer 2008.

• wikipedia: modular form

Revised on November 6, 2013 00:53:23 by Urs Schreiber (77.251.114.72)