Elliptic curve

An elliptic curve over a field $K$ is a smooth projective curve of genus one equipped with a specified $K$-rational point $O$. The chord-and-tangent law makes its rational points an abelian group with identity $O$.

A Weierstrass equation presents an elliptic curve aswith nonzero discriminant, together with the point at infinity $O$.

Over a discretely valued field, a minimal Weierstrass equation has integral coefficients and has least possible discriminant valuation among integral Weierstrass equations related by admissible changes of variables.

Reducing a minimal integral Weierstrass equation modulo the maximal ideal gives a projective cubic over the residue field. Reducing primitive projective coordinates defines the reduction map on rational points.

An inflection point of a smooth plane cubic is a point where its tangent line has intersection multiplicity three. Once one inflection point is chosen as the identity, the nine geometric inflection points are exactly the 3-torsion points.

A one-dimensional commutative formal group law over a ring $R$ is a series $F(X,Y)\in R[[X,Y]]$ satisfying the identity, commutativity, and associativity axioms. A homomorphism from $F$ to $G$ is a series $\theta (T)\in TR[[T]]$ satisfying $\theta (F(X,Y))=G(\theta (X),\theta (Y))$.

Over a characteristic-zero field, a one-dimensional commutative formal group law has a unique strict isomorphism to the additive formal group, called its formal logarithm. It satisfies ${\log }_F(F(X,Y))={\log }_F(X)+{\log }_F(Y)$.

The local parameter $z=-x/y$ at the identity of a Weierstrass curve expresses addition as a formal group law. Over a local field it describes points reducing to the identity.

For an elliptic curve over a nonarchimedean local field, $E_0$ consists of points with nonsingular reduction, $E_1$ is the kernel of reduction, and $E_r$ for $r\ge 1$ consists of points whose formal parameter has valuation at least $r$. The successive quotients for $r\ge 1$ are copies of the additive residue-field group.

An isogeny is a nonconstant morphism of elliptic curves preserving their identity points. It is a finite surjective group homomorphism.

For an isogeny $\varphi :E\to E^{\prime }$, its kernel is denoted $E[\varphi ]$. In characteristic zero its cardinality equals $\mathrm{deg}\varphi$.

The dual of an isogeny $\varphi :E\to E^{\prime }$ of degree $n$ is the unique isogeny $\hat{\varphi }:E^{\prime }\to E$ satisfying

For $\varphi \in \mathrm{End}(E)$, its trace is the integer characterized byEquivalently, $\mathrm{tr}(\varphi )=1+\mathrm{deg}\varphi -\mathrm{deg}(1-\varphi )$.

For an elliptic curve over ${\mathbb{F}}_q$, the Frobenius isogeny sends $(x,y)$ to $(x^q,y^q)$. If $a=q+1-\#E({\mathbb{F}}_q)$, then

If $\alpha ,\beta$ are the roots of $T^2-aT+q$, where $a=q+1-\#E({\mathbb{F}}_q)$, then

For an elliptic curve over ${\mathbb{F}}_q$,

If $a=q+1-\#E({\mathbb{F}}_q)$, then

The $n$-torsion subgroup $E[n]$ is the kernel of multiplication by $n$. When the characteristic does not divide $n$, it is isomorphic over an algebraic closure to $(\mathbb{Z}/n\mathbb{Z}{)}^2$.

The $n$-division field is obtained by adjoining to $K$ the coordinates of every point of $E[n]$. Over a finite field its degree is the order of Frobenius acting on $E[n]$.

The Weil pairing is a bilinear, alternating, nondegenerate pairingFor an isogeny $\varphi$ and its dual, it satisfies $e_n(\varphi P,Q)=e_n(P,\hat{\varphi }Q)$ whenever both sides are defined.

For coprime integers $x,y$, the multiplicative height of $[x:y]\in {\mathbb{P}}^1(\mathbb{Q})$ is $H([x:y])=\max \{|x|,|y|\}$; its logarithmic height is $h=\log H$.

The canonical height isIt is a quadratic form, vanishes on torsion points, and satisfies $\hat{h}(\varphi P)=\mathrm{deg}(\varphi )\hat{h}(P)$ for every isogeny $\varphi$.

The regulator is the determinant of the canonical-height pairing on a basis of the free part of the Mordell-Weil group. A unimodular basis change leaves it invariant.

For a number field $K$, the Mordell-Weil theorem says that $E(K)$ is a finitely generated abelian group.

The multiplication-by-$n$ sequence gives an injective connecting mapLocal conditions restrict its image to the finite $n$-Selmer group.

When ${\mu }_n\subset K$, the Kummer pairing sends $\sigma \in \mathrm{Gal}(L/K)$ and $[x]\in K^{\ast }/(K^{\ast }{)}^n$ to $\sigma (\sqrt[n]{x})/\sqrt[n]{x}\in {\mu }_n$. Its elliptic analogue sends $\sigma$ and $[P]\in E(K)/nE(K)$ to $\sigma (Q)-Q\in E[n]$ for $nQ=P$.

For a number field $K$ and a finite set $S$ of finite primes,

For $E:y^2=x^3+a{x}^2+bx$ and its two-isogenous curve $E^{\prime }:y^2=x^3-2a{x}^2+(a^2-4b)x$, square-class maps on $E(\mathbb{Q})$ and $E^{\prime }(\mathbb{Q})$ determine the Mordell-Weil rank.