A modular form of weight \(k \in \mathbb{Z}\) is a holomorphic function \(f\) defined on the complex upper half plane \(\{\tau \in \mathbb{C} : \mathrm{Im}(\tau) > 0\}\) satisfying the transformation law \[f\left(\frac{a \tau + b}{c \tau + d}\right) = (c \tau + d)^k f(\tau)\] for all \(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z})\) and such that \(f\) is bounded as \(\mathrm{Im}(\tau) \to \infty\).
This simple definition leads to an incredibly rich mathematical world connecting to many other fields of mathematics, especially number theory. Here is a general-interest article about them. We will explore the properties of modular forms using second-year complex analysis and explore some of these connections. We will also investigate the ways in which mathematicians have tried to generalise modular forms.
Prerequisites:
Complex Analysis II.
Corequisites:
None.
Sources:
- Serre, "A Course in Arithmetic", chapter VII
- Bruinier, van der Geer, Harder, Zagier, "The 1-2-3 of Modular Forms".
- Diamond and Shurman, "A First Course in Modular Forms".
- The LMFDB is a rich database of examples of modular forms.