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.

Elliptic curves over the complex numbers

one dimensional complex tori and lattices in the complex numbers

An elliptic curve over the complex numbers is a Riemann surface of genus 1, or a two dimensional torus over the real numbers. The universal cover of this torus, as a complex manifold, is the complex line $\mathbb {C}$. Hence the elliptic curve a isomorphic to quotient of the complex numbers by a some lattice; moreover two elliptic curves are isomorphic if and only the two corresponding lattices are isomorphic. Hence the moduli of elliptic curves over the complex numbers is identified the moduli of lattices in $\mathbb {C}$ up to homothety. For each homothety class there is a lattice such that one of the points of the lattice is 1, and the other is some point $\tau$ in the upper half plane ${\mathcal {H}}$. Hence the moduli of lattices in $\mathbb {C}$ is the quotient ${\mathcal {H}}/PSL_{2}(\mathbb {Z} )$, where $PSL_{2}(\mathbb {Z} )$ acts on the upper half plane via mobius transformations. The standard fundamental domain for this action is the set: $\{\tau |-{\frac {1}{2}}<Im(\tau )\leq {\frac {1}{2}},|\tau |\geq 1,Im(\tau )<0\Rightarrow |\tau |>1\}$.

Theta functions

For the main article see Theta function

Selected References

Further reading

• Joseph H. Silverman, John Tate; "Rational Points on Elliptic Curves". ISBN 0387978259.
• Joseph H. Silverman "The Arithmetic of Elliptic Curves" ISBN 0387962034

Selected external links

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