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.

Curves of genus 1 as smooth plane cubics

If $f(x,y,z)$ is a homogenous degree 3 (also called "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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle genus(E)-1=-2+4/2=0}

On the other hand, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is a smooth algebraic curve of genus 1, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p,q,r} are points on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} , then by the Riemann-Roch formula we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(O_C(p+q+r))=3-(1-1)-h^0(-(p+q+r))=3.} . Choosing a basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_0,g_1,g_2} to the three dimensional vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0(O_C(p+q+r))=\{g:C\to\mathbb{P}^1} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is algebraic and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^{-1}(\infty)=\{p,q,r\}\}} , the map given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s\in C\mapsto (g_0(s):g_1(s):g_2(s))\in\mathbb{P}^2} is an embedding.

The group operation on a pointed smooth plane cubic

Addition on cubic with a marked point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O}

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} be as above, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} point on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} are two points on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p*q:=\overline{pq}\cap E\setminus\{p,q\},} where if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=q} we take the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p(E)} instead, and the intersection is to be understood with multiplicities. The addition on the elliptic curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

Weierstrass forms

Suppose that the cubic curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} admits a flex defined over K, that is, a line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} which is tri-tangent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} at a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} : this will happen, for example, if the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is algebraically closed). In this case there is a change of coordinates on the projective plane which takes the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} to the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{z=0\}} and the point $p$ to the point $(0:1:0)$: we may thus assume that the only terms in the cubic polynomial $f$ which include $y$, are $y^{2}z,xyz,yz^{2}$. The equation can then be put in generalised Weierstrass form

$Y^{2}+a_{1}XY+a_{3}Y=X^{3}+a_{2}X^{2}+a_{4}X+a_{6}.$

If the characteristic of $K$ is not 2 or 3 then by another change of coordinates, the cubic polynomial can be changed to the form

$Y^{2}=X^{3}-27c_{4}X-54c_{6}.$

In this case the discriminant of the cubic polynomial on the right hand side of the equation is given by $\Delta =(c_{4}^{3}-c_{6}^{2})/1728$, and is non-zero because the curve is non-singular. The $j$ invariant of the curve $E$ is defined to be $c_{4}^{3}/\Delta$. Two elliptic curves are isomorphic over an algebraically closed field if and only if they have the same $j$ invariant.

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 is isomorphic to a quotient of the complex numbers by 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 with 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 ${\mathcal {H}}$.

Each triangular region is a free regular set of ${\mathcal {H}}/SL_{2}(\mathbb {Z} )$;; the grey one (with the third point of the triangle at infinity) is the canonical fundamental domain.

Hence the moduli of lattices in $\mathbb {C}$ is the quotient ${\mathcal {H}}/PSL_{2}(\mathbb {Z} )$, where a group element

$\left({\begin{matrix}a&b\\c&d\end{matrix}}\right)\in SL_{2}(\mathbb {Z} )$ acts on the upper half plane via the mobius transformation $z\mapsto {\frac {az+b}{cz+d}}$. The standard fundamental domain for this action is the set: $\{\tau |-{\frac {1}{2}}\leq Im(\tau )\leq {\frac {1}{2}},|\tau |\geq 1\}$.

Modular forms

For the main article see Modular forms Modular forms are functions on the upper half plane, such that for any $\gamma =\left({\begin{matrix}a&b\\c&d\end{matrix}}\right)\in SL_{2}(\mathbb {Z} )$ we have $f(\gamma (\tau ))=(c\tau +d)^{k}f(\tau )$ for some $k$ which is called the "weight" of the form.

Theta functions

For the main article see Theta function

Weierstrass's $\wp$ function

Let $\Lambda =\omega _{1}\mathbb {Z} +\omega _{2}\mathbb {Z}$ be a lattice. The Weirstrass $\wp$-function is the absolutely convergent series $\wp (z,\Lambda )={\frac {1}{z^{2}}}+\sum _{\omega \in \Lambda }'{\frac {1}{(z-\omega )^{2}}}-{\frac {1}{\omega ^{2}}}$ where the sum is taken over all nonzero lattice points. It is an elliptic function having poles of order two at each lattice point.

Elliptic curves over number fields

Let K be an algebraic number field, a finite extension of Q, and E an elliptic curve defined over K. Then E(K), the points of E with coordinates in K, is an abelian group. The structure of this group is determined by the Mordell-Weil theorem, which states that E(K) is finitely generated. By the fundamental theorem of finitely generated abelian groups we have

$E(K)\cong \mathbf {Z} ^{r}\oplus T,\,$

where the torsion-free part has finite rank r, and the torsion group T is finite.

It is not known whether the rank of an elliptic curve over Q is bounded. The elliptic curve

$y^{2}+xy+y=x^{3}-x^{2}-20067762415575526585033208209338542750930230312178956502x+34481611795030556467032985690390720374855944359319180361266008296291939448732243429$

has rank at least 28, due to Noam Elkies .

The torsion group of a curve over Q is determined by Mazur's theorem; over a general number field K a result of Merel shows that the torsion group is bounded in terms of the degree of K.

The rank of an elliptic curve over a number field is related to the L-function of the curve by the Birch-Swinnerton-Dyer conjectures.

Mordell-Weil theorem

The proof of the Mordell-Weil theorem combines two main parts. The "weak Mordell-Weil theorem" states that the quotient $E(K)/2E(K)$ is finite: this is combined with an argument involving the height function.

The theorem also applies to an abelian variety A of higher dimension over a number field. The Lang-Néron theorem implies that A(K) is finitely generated when K is finitely generated over its prime field.

Mazur's theorem

Mazur's theorem shows that the torsion subgroup of an elliptic curve over Q must be one of the following

$C_{1};C_{2};C_{3};C_{4};C_{5};C_{6};C_{7};C_{8};C_{9};C_{10};C_{12};\,$

$C_{2}\times C_{2};C_{2}\times C_{4};C_{2}\times C_{6};C_{2}\times C_{8}.\,$

Each of these torsion structures is parametrizable.