= Elliptic curve group law from Riemann-Roch
{title2=$E(k)\cong\operatorname{Pic}^0(E)$}
= Elliptic curve group law
{synonym}
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 $P+Q\sim R+O$.
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