I am trying to show by hand, without using the Riemann-Roch Theorem, that in $\Bbb C_{\infty}$ we will have that $H^1(0)=0$ and $H^1(-p)=0$, but I am having some trouble constructing the functions that are related to the Laurent series, so I guess this is more a complex analysis question that if I have a Laurent series that is like $a_n\frac{1}{z^n}+\ldots+a_0\frac{1}{z}$ at point $p$ and $b_k\frac{1}{z^k}+\ldots+b_m\frac{1}{z^m}$ at point $q$ how can i construct a meromorphic function that will have these laurent series at these points and be holomorphic elesewhere and so on ? I am bit confused on how to make things work.
Also I am trying do another exercise that it's kind of the same idea that I cant find the desired functions, it goes like this
Let $X$ be the complex torus, fix a finite number of points $p_i \in X$ with local coordinate $z_i$ at $p_i$, consider the laurent divisor $Z=\sum\limits_{i}c_iz_i^{-1}.p_i$, show that $Z=\alpha_0(f)$ for some global meromorphic function $f$ iff $\sum\limits_i c_i=0$.
I was able to do the first implication using the residue theorem applied to $fdz$, now for the second one I am trying to construct the function by taking other points $q_j$ such that $\sum\limits_j q_j - \sum\limits_i p_i \in \Bbb Z$ and then using the ratio of theta-translated functions $\frac{\prod_j \vartheta^{q_j}(z)}{\prod_i \vartheta^{p_i}(z)}$ but I dont know how I am going to make the functions get the residue $c_i$ at the point $p_i$, I have to change something from that function but I dont know what.
So any tips or help would be aprecciated. Thanks in advance.