# Contents

## Idea

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

Modular forms appear as

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)

note take $A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ then we get that $f\left(\tau +1\right)=f\left(\tau \right)$

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 ${a}_{k}\in ℤ$

by this definition, modular forms are not really functions on the upper half plane, but function on a moduli space of tori.

• 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 August 25, 2012 23:16:18 by Marc Olschok (78.53.188.81)