Elliptic curve group law from Riemann-Roch ($E(k)\cong {\mathrm{Pic}}^0(E)$)

Fix $O\in E(k)$. On a genus-one smooth projective curve, every degree-one divisor class has a unique effective point representative by the Riemann-Roch theorem: its section space has dimension one, since the complementary canonical-divisor degree is negative. Thus $P\mapsto [P-O]$ is a bijection onto the degree-zero Picard group of a curve. Transporting divisor-class addition proves the group axioms, with identity $O$, and characterizes $P+Q$ by $P+Q\sim R+O$.