Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 29 / 2 / i / Solution

The projective completion has identity $O=[0:1:0]$. For this Weierstrass equation of an elliptic curve, negation isFor distinct affine points $P,Q$, draw their chord; for $P=Q$, use the tangent with its intersection multiplicity. If the third intersection is $R$, define $P\oplus Q=-R$. A vertical chord pairs $P$ with $-P$ and gives their sum $O$; also $P\oplus O=P$. A vertical tangent at a point satisfying $2y+1=0$ gives $2P=O$.

For a nonvertical line $y=mx+b$, substitution into the cubic givesThus if $P=(x_1,y_1)$ and $Q=(x_2,y_2)$, the sum has coordinatesHere $m=(y_2-y_1)/(x_2-x_1)$ for a chord, and implicit differentiation givesfor a nonvertical tangent. These formulas include the repeated-root case. Associativity can be seen without a lengthy coordinate calculation: a line cuts the cubic in a divisor equivalent to $3(O)$, and $P\mapsto [(P)-(O)]$ identifies the curve with its degree-zero divisor class group. The chord-and-tangent operation becomes addition of divisor classes, hence is associative.