Say $\tau $ be a complex number with $\Im(\tau)>0$.Then a meromorphic function $f$ on the torus $M$ can be thought of as a meromorphic function on the plane with
$f(z+1)=f(z)$ and $f(z+\tau)=f(z)$ $\forall z\in \mathbb{C}$.
The Weierstrass $\wp$-function is defined as follows:
$\wp(z)=\frac{1}{z^2}+\sum\limits_{(m,n)\in \Lambda\backslash (0,0)}(\frac{1}{{(z-n-m\tau)}^2}-\frac{1}{{(n+m\tau)}^2})$ (where $\Lambda $ is the lattice generated by $1$ and $\tau$)
I am stuck with the following question:
Any $f\in L[P^{-1}Q^{-1}]\backslash \mathbb{C}$ meromorphic on the torus can be written as
$f=\beta\circ\wp\circ\alpha$ where $\beta$ is an automorphism of the Riemann sphere and $\alpha$ is an automorphism of the torus.
Now if the $\wp$-function sends both $P$ and $Q$ to the same $a$ on $\mathbb{C}\cup\{\infty\}$ we can take $\beta$ to be $\frac{1}{z-a}$ (with the adjustments of a scaling and a constant perhaps) as $L[P^{-1}Q^{-1}]$ is two-dimensional .
But I don't know how to proceed if $P$ and $Q$ are mapped to two different points .Perhaps, we can take $a\in \mathbb{C}\cup\{\infty\}$ with $|\wp^{-1}\{a\}|=2$ and then find an automorphism of the torus $T$ that takes the two points in the inverse image of $a$ (via the $\wp$ ) to $P$ and $Q$ respectively.But I don't know how to find such an automorphism of $T$ ( I don't know how the automorphism group of $T$ operates).
Please help me on this.