Let $T$ be the complex tore from the lattice $(1, \tau)$ where $im(\tau)>0$.
How to prove the existence of a meromorphic function on $T$, with divisor $(0) + (\frac{1}{2}) - 2 (\frac{\tau}{2})$ ? (i.e with simple zeros at 0 and $\frac{1}{2}$ and a double pole at $\frac{\tau}{2}$).
I have troubles using Riemann Roch for the question I ask because it gives the dimension of the space based on an inequality of divisors.
Clue
If it can help, the question above is the 2nd question of an exercise, the first one being: what are the holomorphic 1-forms on $T$?
My answer for this preliminary question: if $\omega$ is the Weirstrass invariant 1-form, then all 1-forms are $f.\omega$ with $f$ meromophic. $f$ must be of the form $g(\rho) + \rho'h(\rho)$ and since we don't want any poles this leaves only constants: all holomorphic 1-forms are $k \omega, k \in \mathbb{C}$. So $l(K)=1$ (from Riemann Roch)
But then I don't know how this helps for finding principal divisors of $(0) + (\frac{1}{2}) - 2 (\frac{\tau}{2})$
A theorem says that if $x_i$ are the zeros or poles of a function with orders $n_i$, then $\sum n_i x_i \in \Lambda$ which is not the case here.