= Associativity of the chord-and-tangent law via divisor classes
For a smooth <genus one curve> with origin $O$, the map $P\mapsto[(P)-(O)]$ identifies geometric points with the <degree-zero Picard group of a curve>. <Riemann-Roch theorem> gives a unique effective representative for each divisor class of degree one: its Riemann-Roch space has dimension one. A line cuts a plane cubic in a divisor linearly equivalent to $3(O)$ when $O$ is the Weierstrass flex. The two successive line intersections in the <chord-and-tangent group law> therefore give addition of the corresponding divisor classes. Associativity follows by injectivity of the identification and associativity of divisor-class addition, including tangent intersection multiplicities.
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