About the quotient group of degree zero divisors on $C$ by the principal divisors on $C$

Question

Let $C$ be an elliptic curve with distinguished point $O$. My question is about a mathematical desription of this set denoted by $Pic(C)$ which is the quotient group of degree zero divisors on $C$ by the principal divisors on $C$.

Answer 1

What you are asking about is usually denoted $Pic^0(C)$. It is a standard result that this group is isomorphic, as an abstract group, to the elliptic curve $C$ itself. The isomorphism is obtained by taking a divisor class $\sum n_i [P_i]$ and mapping it to the point $\sum n_i P_i$ on $C$ (the sum here being with respect to the group law on $C$).