Suppose $a,b,c,d$ are integers with $ad-bc\ne 0$ and write the torus $T$ as $\{(x,y)\in \mathbb C^2: 1=|x|=|y|\}.$ Consider the map $p: T\to T$ defined by $(x,y)\to (x^ay^b,x^cy^d)$.
I am able to prove that $p$ is a covering space.
I want to now find the cardinality $|p^{-1}(x_0)|$ for each $x_0\in T$.
Could someone please explain how this works?
Let's use $(u, v)$ for the complex coordinates, so that we can write $$ u = e^{ix},\quad v = e^{iy}. $$ Since $a$, $b$, $c$, and $d$ are integers, the map $p$ sends $(u, v) = (e^{ix}, e^{iy})$ to $$ (u^{a} v^{b}, u^{c} v^{d}) = (e^{(ax + by)i}, e^{(cx + dy)i}). $$ It should be apparent that the lift $\hat{f}$ of $p$ is as in the hint.
To count the number of sheets in the covering $p$, consider the image of some fundamental domain, such as the unit square $[0, 1] \times [0, 1]$ under $\hat{f}$, and ask "how many unit squares" the image covers (i.e., what is its area?).
The number of sheets is: