Let $X$ be a compact Riemann Surface of genus 1. Let $Cl_0(X) := \frac{Div(X)}{PDiv(X)}$, where $PDiv(X)$ is the subgroup of principal divisors on $X$. Let $P \in X$ be a fixed point. We have a bijection
$i_p : X \rightarrow Cl_0(X)$
given by $X \in Q \rightarrow [Q-P]$.
Using this bijection one can pullback the (commutative)group structure of $Cl_0(X)$ to X as follows. Let $\mu$ be the multiplication map $\mu : X \times X \rightarrow X$ be defined by
$\mu(Q_1,Q_2) = i_p^{-1}(i_p(Q_1) + i_p(Q_2))$.
Now my question is how to prove that this group structure makes $X$ into a complex Lie group. Thanks.