Question about divisors Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of finite order (say, $k$).
How do I find the dimension and a basis of $l(nO)$ and $l(nO-mA)$?
For $l(nO)$ its dimension is $n$ and I think $\{1,\wp,\wp',\ldots,\wp^{(n-2)}\}$ would be a basis.
For $l(nO-mA)$ I am unsure of the dimension (is it $\deg(nO-mA)=n-m$?). Also, I do not know how to find a basis?
 A: I interpret the question as to imply that $A$ is a point of two-torsion on the elliptic curve, i.e. $A\neq 0$ and $2A=0$ in the underlying group.
a) We have by Riemann-Roch:     
$ l(nO)=
\left  \{\begin  {align} 
0 \; \operatorname {if} \; n\lt 0\\
1 \;\operatorname {if} \; n=0\\
n \;\operatorname {if} \; n\gt 0
\end {align} 
\right. \ $ 
b) Using Riemann-Roch again we get: 
$ l(nO-mA)=
\left  \{\begin  {align} 
0 \; \operatorname {if} \; n-m\lt 0\\
n-m \;\operatorname {if} \;n-m\gt 0
\end {align} 
\right. \ $
If $n=m$ , we have by moreover invoking  Abel's theorem:
  $ l(nO-nA)=
\left  \{\begin  {align} 
1 \; \operatorname {if} \; n\in 2\mathbb Z\\
0 \; \operatorname {if} \; n\notin 2\mathbb Z
\end {align} 
\right. \ $   
Edit
In the case the OP asks about in the comments where  $A\neq 0$ is an arbitrary point of the elliptic curve $C$ , the last case of my answer should be modified to :  
If $n=m$ , we have by moreover invoking  Abel's theorem:
  $ l(nO-nA)=
\left  \{\begin  {align} 
1 \; \operatorname {if} \; nA=0 \in C\\
0 \; \operatorname {if} \; nA\neq 0 \in C
\end {align} 
\right. \ $  
