Difference between revisions of "Modular form"
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| − | 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: | + | 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:<br /> |
| − | + | 1. $f$ is [[holomorphic]] on $\mathbb{H}$, <br /> | |
| − | + | 2. for any $z \in \mathbb{H}$ and $\begin{bmatrix} a&b \\ c&d \end{bmatrix} \in \mathrm{SL}(2,\mathbb{Z})$, | |
$$f \left( \dfrac{az+b}{cz+d} \right) = (cz+d)^k f(z),$$ | $$f \left( \dfrac{az+b}{cz+d} \right) = (cz+d)^k f(z),$$ | ||
| − | and | + | and <br /> |
| − | + | 3. $f$ is [[holomorphic at the cusp]]. | |
=Properties= | =Properties= | ||
Latest revision as of 03:20, 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:
1. $f$ is holomorphic on $\mathbb{H}$,
2. 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
3. $f$ is holomorphic at the cusp.
