Consider a function f \in \overline{K}(C)^{*}, then we can look at the formal sum div(f) = \sum_{P \in C}{\mathrm{ord}_{P}(f)[P]}.
The Abel-Jacobi theorem states that a divisor D = \sum_{P \in E}{c_P [P]} is principal if and only if D has degree 0 and \sum_{P \in E}{c_P P} = O under the usual addition law for points on cubic curves.