Difference between revisions of "Modular form"
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Revision as of 03:19, 26 February 2018
A modular form of weight $k$ for $\mathrm{SL}(2,\mathbb{Z})$ is a function $f \colon \mathbb{H} \rightarrow \mathbb{C}$, where $\mathbb{H}$ is a the upper half-plane that satisfies three conditions:
- $f$ is holomorphic on $\mathbb{H}$,
- for any $z \in \mathbb{H}$ and $\begin{bmatrix} a&b \\ c&d \end{bmatrix} \in \mathrm{SL}(2,\mathbb{Z})$,
$$f \left( \dfrac{az+b}{cz+d} \right) = (cz+d)^k f(z),$$ and
- $f$ is holomorphic at the cusp
