Dimension of $L(D)$ space

Question

Let $X$ be a compact Riemann surface such that $g(X)=1$, and $p \in X$. Let's consider the divisor $D = n[p]$, where $n$ is n is natural. What's the dimension of $L(D)$ ? Thanks in advance for your answers.

Answer 1

Because $g(X)=1$, $X$ is a complex torus $X=\mathbb{C}/\Lambda$. You should consult Miranda's Algebraic Curves and Riemann Surfaces p.150, where a proposition implies that (with slight adaptation): for $D=n\cdot p$, $n\in \mathbb{Z}$

$(1)$ if $n>0$, $\dim L(D)=\deg(D)=n$

$(2)$ if $n=0$, $\dim L(D)=1$

$(3)$ if $n<0$, $L(D)=\{0\}$, and hence $\dim L(D)=0$.