Editing Modular form (section)

=== Standard definition ===
A modular form of weight <math>k</math> for the [[modular group]]
:<math>\text{SL}(2, \Z) = \left \{ \left. \begin{pmatrix}a & b \\ c & d \end{pmatrix}  \right | a, b, c, d \in \Z,\ ad-bc = 1 \right \}</math>
is a function <math>f</math> on the [[upper half-plane]] <math>\mathcal{H}=\{z\in\C\mid \operatorname{Im}(z)>0\}</math> satisfying the following three conditions: 
# <math>f</math> is [[holomorphic function|holomorphic]] on <math>\mathcal{H}</math>. 
# For any <math>z\in\mathcal{H}</math> and any matrix in <math>\text{SL}(2, \Z)</math>, we have
#:<math> f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z)</math>.
# <math>f</math> is bounded as <math>\operatorname{Im}(z)\to\infty</math>.

Remarks: 
* The weight <math>k</math> is typically a positive integer.
* For odd <math>k</math>, only the zero function can satisfy the second condition.
* The third condition is also phrased by saying that <math>f</math> is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some <math> M, D > 0 </math> such that <math> \operatorname{Im}(z) > M \implies |f(z)| < D </math>, meaning <math>f</math> is bounded above some horizontal line.
* The second condition for 
::<math>S = \begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad T = \begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix}</math> 
:reads
::<math>f\left(-\frac{1}{z}\right) = z^k f(z), \qquad f(z + 1) = f(z)</math>
:respectively. Since <math>S</math> and <math>T</math> [[generating set of a group|generate]] the group <math>\text{SL}(2, \Z)</math>, the second condition above is equivalent to these two equations. 
* Since <math>f(z+1)=f(z)</math>, modular forms are [[periodic function]]s with period {{math|1}}, and thus have a [[Fourier series]].