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Following wikipedia, a modular form of weight $k$ for the modular group $\mathrm{SL}_2(\mathbb{Z})$ is a complex-valued function  $f$  on the upper half-plane $H = \{z\in\mathbb{C}\colon Im(z)>0\}$ satisfying the following:

  1. $f$  is a holomorphic function on $H$.

  2. For any $z\in\mathbb{H}$, and any matrix $\begin{bmatrix} a & b\\c & d\end{bmatrix}$ in $\mathrm{SL}_2(\mathbb{Z})$, we have: $$f\Big{(} \frac{az+b}{cz+d} \Big{)}=(cz+d)^kf(z)$$

    3. $f$  is required to be holomorphic as $z\rightarrow i\infty$.

I am just entring the world of modular forms. I have basic questions here:

(1) In condition 2, if we remove $(cz+d)^k$ then the condition will say that $f$ is invariant under $\mathrm{SL}_2(\mathbb{Z})$, and the study of object invariant under any nice group are interesting always; then why this term $(cz+d)^k$ introduced in second condition?

(2) Where can I find the origin of the theory of modular forms?

Mathmo123
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    As a partial answer to your first question, removing that factor is the same as taking $k=0$. It turns out that the only holomorphic functions which are invariant under $\mathrm{SL}_2(\mathbb Z)$ and satisfy condition $3$ are the constant functions. – Mathmo123 Mar 31 '16 at 04:07
  • You should also look here for some answers to your second question. http://math.stackexchange.com/questions/325364/intuition-for-the-importance-of-modular-forms – Mathmo123 Mar 31 '16 at 04:11

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Modular forms are really functions of lattices in $\mathbb{C}:$ these are additive subgroups of the form $\mathbb{Z}w_1 \oplus \mathbb{Z} w_2$, where $w_1,w_2 \in \mathbb{C}$ are $\mathbb{R}$-linearly independent. If we only look at lattices up to conformal equivalence (scaling and rotating), we can assume the lattice is given by $$\mathbb{Z} \oplus \mathbb{Z}\tau, \;\; \tau = \frac{w_2}{w_1} \in \mathbb{H}.$$ Two $\tau,\tau'$ define the same lattice if and only if there is a change-of-basis matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z})$ with $$\mathbb{Z} \oplus \mathbb{Z} \tau' = \mathbb{Z} (c \tau + d) \oplus \mathbb{Z} (a \tau + b).$$

A modular function of weight $k$ is a holomorphic function $f(\tau) = f(\mathbb{Z} \oplus \mathbb{Z}\tau) = f(\Lambda)$ that scales by the factor $C^{-k}$ whenever $\Lambda$ is rescaled by $C$. In particular, $$f(\tau) = f(\mathbb{Z} \oplus \mathbb{Z}\tau) = f(\mathbb{Z}(c\tau + d) \oplus \mathbb{Z} (a \tau + b)) = (c \tau + d)^{-k} f\Big(\mathbb{Z} \oplus \mathbb{Z} \frac{a \tau + b}{c \tau + d} \Big) = (c \tau + d)^{-k} f\Big( \frac{a \tau + b}{c \tau + d} \Big)$$ for any $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

Some modular functions do not change under scaling this way (so $k=0$). The most famous is Klein's j-invariant (this is not a modular form, though, since it is not holomorphic as $\mathrm{Im}(\tau) \rightarrow \infty$). However the first examples of modular functions arise from lattice sums relating to $$\wp(z,\Lambda) = \frac{1}{z^2} + \sum_{(m,n) \in \Lambda \backslash \{(0,0)\}} \Big( \frac{1}{(z+m+n)^2} - \frac{1}{(m+n)^2} \Big) = \frac{1}{z^2}\Big( 1 + \sum_{n=2}^{\infty} (2n-1) G_{2n}(\Lambda) z^{2n} \Big);$$ clearly $\rho(Cz,C\Lambda) = C^{-2} \rho(z,\Lambda),$ and comparing coefficients implies that $G_{2n}$ must scale $\Lambda$ by $C^{-2n}.$

This connection with lattices (and by extension, the quotients of $\mathbb{C}$ by lattices, which are elliptic curves) is probably the original motivation to study modular forms. Since then, people have turned up many other surprising situations where they arise.

user327376
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