Elliptic curve

An elliptic curve over a field $K$ is a one dimensional Abelian variety over $K$. Alternatively it is a smooth algebraic curve of genus one together with marked point - the identity element.

Curves of genus 1 as smooth plane cubics

If $f(x,y,z)$ is a homogenous cubic polynomial in three variables, such that at no point $(x:y:z)\in \mathbb {P} ^{2}$ all the three derivatives of f are simultaneously zero, then the Null set $E:=\{(x:y:z)\in \mathbb {P} ^{2}|f(x,y,z)=0\}\subset \mathbb {P} ^{2}$ is a smooth curve of genus 1. Smoothness follows from the condition on derivatives, and the genus can be computed in various ways; e.g.:

• Let $H$ be the class of line in the Picard group $Pic(P^{2})$, then $E$ is rationally equivalent to $3H$. Then by the adjunction formula we have $\#K_{E}=(K_{\mathbb {P} ^{2}}+[E])[E]=(-3H+3H)3H=0$.
• By the genus degree formula for plane curves we see that $genus(E)=(3-1)(3-2)/2=1$
• If we choose a point $p\in E$ and a line $L\subset \mathbb {P} ^{2}$ such that $p\not \in L$, we may project $E$ to $L$ by sending a point $q\in E$ to the intersection point ${\overline {pq}}\cap L$ (if $p=q$ take the line $T_{p}(E)$ instead of the line ${\overline {pq}}$). This is a double cover of a line with four ramification points. Hence by the Riemann-Hurwitz formula $genus(E)-1=-2+4/2=0$

On the other hand, if $C$ is a smooth algebraic curve of genus 1, and $p,q,r$ are points on $C$, then by the Riemann-Roch formula we have $h^{0}(O_{C}(p+q+r))=3-(1-1)-h^{0}(-(p+q+r))=3.$

Hence the complete linear system $|O_{C}(p+q+r)|$ is two dimensional, and the map from $C$ to the dual linear system is an embedding.

The group operation on a pointed smooth plane cubic

Let $E$ be as above, and $O$ point on $E$. If $p$ and $q$ are two points on $E$ we set $p*q:={\overline {pq}}\cap E\setminus \{p,q\},$ where if $p=q$ we take the line $T_{p}(E)$ instead, and the intersection is to be understood with multiplicities. The addition on he elliptic curve $E$ is defined as $p+q:=O*(p*q)$. Both the commutativity and the existence of inverse follow from the definition. The proof of the associativity of this operation is more delicate.