Interpretations of the automorphy factor

Question

Let $z \in{\mathbb{H}} = \{ z \in{\mathbb{C}: \mathrm{Im } (z) > 0} \}$, $k \in{\mathbb{Z}}$, and $\begin{pmatrix}a&b \\ c&d \end{pmatrix} \in{SL_2(\mathbb{Z})}$. The automorphy factor of weight $k$ may be defined as $$ j_k(z,\begin{pmatrix}a&b \\ c&d \end{pmatrix}) = (cz+d)^{-k} $$

Examples:

1. $j_1(z,\begin{pmatrix}0&-1 \\ 1&0 \end{pmatrix}) = \frac{1}{z}$, which is complex inversion, while $\begin{pmatrix}0&-1 \\ 1&0 \end{pmatrix} z = \frac{0z-1}{z+0} $ which is also an inverion when the action is fractional linear transformation.

2. $j_k(z, \begin{pmatrix}1&n \\ 0&1 \end{pmatrix}) = 1$

3. $j_k(z, \begin{pmatrix}2&1 \\ 1&1 \end{pmatrix}) = \frac{1}{(z+1)^k} $, which is translation then inversion to the $k$ power, while $ \begin{pmatrix}2&1 \\ 1&1 \end{pmatrix} z = \frac{2z+1}{z+1} = 2- \frac{1}{z+1} $ which is translation, inversion, then translation.

There appears to be a connection between what $\begin{pmatrix} a&b\\c&d\\ \end{pmatrix}$ does via fractional linear transformation and how it responds in $j_k$. In view of the above observations: Does $j_k$ pick up "how much inversion" a $T \in{SL_2(\mathbb{Z})}$ does via fractional linear transformation? Is there a geometric interpretation to $j_k$, or a sense of what it measures?

Answer 1

$$\gamma = \pmatrix{a&b\\c&d}, \quad\gamma(z) = \frac{az+b}{cz+d},\quad \gamma'(z)=(cz+d)^{-2}, \quad (\gamma \circ \beta)'(z) = \gamma'\circ \beta(z) \ \beta'(z)$$

If $f$ is holomorphic/meromorphic and weight $k$-invariant for a (finite index) subgroup $\Gamma \le SL_2(\Bbb{Z})$ then the formal expression $f(z)^2 (dz)^k$ is weight $0$ invariant. If $k=2$ then $f(z)dz$ is an holomorphic/meromorphic one-form on the modular curve $\Gamma \setminus \mathbb{H} = \{ \Gamma z,\Im(z)>0\}$.

If $F$ is meromorphic and weight $0$-invariant then $F'$ is weight $2$-invariant.

Once you know one modular form, for example the Eisenstein series $G_{2k}$, you can define all the level $N$ modular forms of weight $2k$ as those functions such that $\frac{f}{G_{2k}}$ is meromorphic on the level $N$-modular curve and with poles at most at the zeros of $G_{2k}$ (ie. what Riemann Roch is about).

Those modular forms appear naturally from the Eisenstein series, the elliptic curves, the Dirichlet series with functional equation and the Poisson summation formula, and they have a rich Hecke operator theory whose eigenvalues correspond to their Fourier coefficients, all this being hardly seen in the field of meromorphic functions on the modular curves.