Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2009 / B / 4 / a

Work with the smooth projective cubic, including its points at infinity. For two points $P,Q$, draw their chord, using the tangent when they coincide, and let $R$ be its third intersection counted with intersection multiplicity. Draw the line through $R$ and $o$, again using the tangent if necessary, and let $S$ be its third intersection. Define $P+Q=S$. When $o$ is a flex, this is the familiar reflection of the first third-intersection point across the Weierstrass involution. The two-line construction also works when the chosen identity is not a flex.

To prove associativity without assuming it in the geometry, use divisor class. A nonsingular plane cubic has genus one. By the Riemann-Roch theorem, every divisor class of degree one contains exactly one effective divisor on an algebraic curve of degree one: for such a divisor on an algebraic curve $D$,and $\mathrm{deg}(K_C-D)=-1$, so $\ell (K_C-D)=0$. Thus $\ell (D)=1$, and the nonzero section is unique up to scalar. Over $K$, a degree-one effective divisor on an algebraic curve defined over $K$ is a $K$-rational point. This also proves uniqueness of that point. In particular the mapis injective and identifies these points with the corresponding degree-zero Picard group of a curve: to recover the point from a class, add $[o]$ and take its unique effective representative. This is the elliptic curve group law from Riemann-Roch.

Let $H$ denote the divisor class cut out by a line. The two lines in the chord-and-tangent group law giveTheir difference gives $P+Q\sim o+S$, henceAddition of divisor class is associative. Therefore both $(P+Q)+R$ and $P+(Q+R)$ map to $[P-o]+[Q-o]+[R-o]$. Injectivity of $\iota$ proves the chord-and-tangent group law is associative. The same argument gives identity $o$, commutativity and inverses, and covers tangencies through the use of intersection multiplicities. All constructions are defined over $K$.