Dimension of space of meromorphic functions on torus with only one pole

Question

I've distilled this down to just the most necessary details in order to avoid cluttering this question. Let $\Gamma$ be some lattice in $\mathbb{C}$. Let, say, $V$ denote the space of meromorphic functions on the torus $\mathbb{C}/\Gamma$ with exactly one (possibly removable) singularity at a certain fixed point $x \in V$ such that, if $f \in V$ then the order of its pole (if there is one) at $x$ is at most $n$. Then apparently the dimension of this space is $n$. I really don't see how - apparently one would have to use the Weierstrass $\wp$-function to understand that....could anybody please provide a bit of guidance?

Answer 1

Without loss of generality assume that the pole is at $z=0$. A function $f \in V$ has a Laurent-series $$ f(z) = \sum_{j=-n}^\infty a_n z^n $$ at $z=0$.

Now consider the Weierstrass $\wp$ function for the same lattice, and its derivatives $\wp', \ldots, \wp^{(n-2)}$. $\wp, \wp', \ldots, \wp^{(n-2)}$ have poles at $z=0$ of order $2, 3, \ldots, n$, respectively. It follows that for suitable constants $c_0, \ldots, c_{n-2}$, $$ g(z) = f(z) - c_0 \wp - c_1 \wp' - \ldots - c_{n-2} \wp^{(n-2)} $$ has at most a simple pole at $z=0$ (and no other poles in the fundamental parallelogram).

And since the sum of all residues of an elliptic function in the fundamental parallelogram is always zero, $g$ has no pole at all and therefore is constant.

It follows that functions in $V$ are exactly the linear combinations $$ c + c_0 \wp + c_1 \wp' + \ldots + c_{n-2} \wp^{(n-2)} $$ with $c, c_0, \ldots, c_{n-2} \in \Bbb C$.