Difference between revisions of "Modular form"

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(Created page with "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 sati...")
 
 
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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:
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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:<br />
#$f$ is [[holomorphic]] on $\mathbb{H}$,
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1. $f$ is [[holomorphic]] on $\mathbb{H}$, <br />
#for any $z \in \mathbb{H}$ and $\begin{bmatrix} a&b \\ c&d \end{bmatrix} \in \mathrm{SL}(2,\mathbb{Z})$,  
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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
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and <br />
#$f$ is [[holomorphic at the cusp]]
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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.

Properties

References