Modular Forms and Möbius transformation
NOTES ON STATISTICS, PROBABILITY and MATHEMATICS
Modular Forms and Möbius transformation:
Mathematics:
A modular form assignes a complex number to each point in a lattice based on a Möbius transformation:
\[g(\small\text{lattice})\to \mathbb C\]
which can be thought of as the function on the basis:
\[g(\omega_1, \omega_2)\to \mathbb C\]
It is useful to make the lattice homogenous, such that re-scaling
\[g(\lambda \omega_1, \lambda\omega_2)=\lambda^{-k}g(\omega_1,\omega_2)\]
where \(k\) is the weight of the modular form.
Choosing a different basis shouldn’t change the output of \(g\). Therefore a change of basis effectuated by a matrix
\[\begin{bmatrix}a & b\\ c & d \end{bmatrix} \in \text{GL}_2 (\mathbb Z) \] such that
\[\begin{bmatrix}a & b\\ c & d \end{bmatrix} \begin{bmatrix}\omega_1\\\omega_2 \end{bmatrix}= \begin{bmatrix}a\omega_1+ b\omega_2\\c\omega_1+d\omega_2 \end{bmatrix}\]
and
\[g(a\omega_1+ b\omega_2, c\omega_1+d\omega_2 ) = g(\omega_1, \omega_2)\]
To work with just one variable, we define
\[f(\tau) = g(\tau,1)=\lambda^{k}g(\lambda\tau,\lambda)\]
We define \[\tau =\frac{\omega_2}{\omega_1}\in \mathcal H\] - i.e the upper-half plane of the complex numbers. So that, given homogeneity,
A lattice \[\Lambda \subseteq \mathbb C\]
such that \[\Lambda= \mathbb Z \omega_1 + \mathbb Z \omega_2 = \omega_1\left( \mathbb Z + \mathbb Z \tau \right)\]
This corresponds to rotating and scaling the lattice as in:
Modular forms are not invariant under transformation, but they are predictable. The transformation of the domain for the modular form is the action of the modular group.See here.
Limiting the \(\text{GL}_2(\mathbb Z)\) to the special linear group \(\text{SL}_2(\mathbb Z)\) acting on points on the complex plane:
\[\text{SL}_2(\mathbb Z)\require{HTML} \style{display: inline-block; transform: rotate(-270deg)}{\circlearrowright} h\in \mathcal H\]
the above operation
\[\begin{bmatrix}a & b\\ c & d \end{bmatrix} \begin{bmatrix}\omega_1\\\omega_2 \end{bmatrix}= \begin{bmatrix}a\omega_1+ b\omega_2\\c\omega_1+d\omega_2 \end{bmatrix}\]
can be written as
\[f(\tau) =g(\tau,1)=g(a\tau + b, c\tau+d)=(c\tau+d)^{-k}g\left(\frac{a\tau+b}{c\tau + d}, 1\right)=(c\tau+d)^{-k}f\left(\frac{a\tau+b}{c\tau + d}\right)\]
A Möbius transformation is (Wikipedia):
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
\[f(z) = \frac{az + b}{cz + d}\]
of one complex variable \(z\); here the coefficients \(a, b, c, d\) are complex numbers satisfying \(ad − bc ≠0.\)
In order to plot this transformation using Cartesian coordinates on a computer platform the real and imaginary components will need to be separated.
Using this post:
\[\begin{align} &\frac {(a_r x - a_i y ) + (a_i x + a_r y) i + b_r + b_i i} {(c_r x - c_i y) + (c_i x + c_r y) i + d_r + d_i i} \\[3ex] = &\frac {(a_r x - a_i y ) + (a_i x + a_r y) i + b_r + b_i i} {(c_r x - c_i y) + (c_i x + c_r y) i + d_r + d_i i} \\[3ex] =&\frac {(a_r x - a_i y+ b_r ) + (a_i x + a_r y + b_i) i } {(c_r x - c_i y+ d_r) + (c_i x + c_r y+ d_i) i} \\[3ex] = & \left( \frac {(a_r x - a_i y+ b_r ) (c_r x - c_i y+ d_r) + (a_i x + a_r y + b_i) (c_i x + c_r y+ d_i) } {(c_r x - c_i y+ d_r)^2 + (c_i x + c_r y+ d_i) ^2} \right) \\[3ex] + & \left( \frac {(a_i x + a_r y + b_i)(c_r x + c_i y+ d_r) -(a_r x - a_i y+ b_r )(c_i x + c_r y+ d_i ) } {(c_r x - c_i y+ d_r)^2 + (c_i x + c_r y+ d_i)^2 } \right) i \end{align}\]
This is implemented here.
See also this answer on Stack exchange.
Or…
NOTE: These are tentative notes on different topics for personal use - expect mistakes and misunderstandings.