nLab
modular form

Idea

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

Modular forms appear as

• as the cohomology ring over the point in the cohomology theory called tmf – “tmf” stands for “topological modular forms”;

• as the partition functions of certain 2-dimensional quantum field theories.

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

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

(complex numbers with strictly positive imaginary part)

such that

1. if $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z})$ acting by $A:\tau \mapsto =\frac{a\tau +b}{c\tau +d}$ we have

   $$f(A(\tau ))=(c\tau +d{)}^{w}f(\tau )$$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 $f(\tau +1)=f(\tau )$

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

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

3. integrality $\tilde{f}(q)={\sum }_{k=-N}^{\mathrm{\infty }}{a}_k\cdot q^k$ then $a_k\in \mathbb{Z}$

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

Links and references

• 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.

• Nora Ganter, Topological modular forms literature list

• wikipedia: modular form