3
$\begingroup$

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?

$\endgroup$
2
  • 3
    $\begingroup$ 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. $\endgroup$
    – Mathmo123
    Mar 31 '16 at 4:07
  • $\begingroup$ You should also look here for some answers to your second question. math.stackexchange.com/questions/325364/… $\endgroup$
    – Mathmo123
    Mar 31 '16 at 4:11
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.