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A non-singular complete
+
{{MSC|14h57|11Gxx,14K15}}
 +
{{TEX|done}}
 +
 
 +
 
 +
An ''elliptic curve'' is a non-singular complete
 
[[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic
 
[[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic
 
curves is the source of a large part of contemporary algebraic
 
curves is the source of a large part of contemporary algebraic
Line 18: Line 22:
 
over an algebraically closed field $k$. Then $X$ is biregularly
 
over an algebraically closed field $k$. Then $X$ is biregularly
 
isomorphic to a plane cubic curve (see
 
isomorphic to a plane cubic curve (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[9]]],
+
{{Cite|La2}},
[[#References|[13]]]). If ${\rm char k} \ne 2,3$, then in the projective plane ${\mathbb P}^2$ there
+
{{Cite|Ta}}). If $\mathop{\mathrm{char}} k \ne 2,3$, then in the projective plane ${\mathbb P}^2$ there
 
is an affine coordinate system in which the equation of $X$ is in
 
is an affine coordinate system in which the equation of $X$ is in
 
normal Weierstrass form:  
 
normal Weierstrass form:  
 
$$y^2=x^3+ax+b$$
 
$$y^2=x^3+ax+b$$
The curve $ $ is non-singular if and
+
The curve $X$ is non-singular if and
only if the polynomial $_$ does not have multiple zeros, that is, if
+
only if the polynomial $x^3+ax+b$ does not have multiple zeros, that is, if
the discriminant $_$. In $_$ the curve (1) has a unique point at
+
the discriminant $\Delta = -16(4a^3+27b^2)\ne 0$. In ${\mathbb P}^2$ the curve (1) has a unique point at
infinity, which is denoted by $_$; $_$ is a point of inflection of
+
infinity, which is denoted by $P_0$; $P_0$ is a point of inflection of
(1), and the tangent at $_$ is the line at infinity. The $_$-invariant
+
(1), and the tangent at $P_0$ is the line at infinity. The $j$-invariant
of an elliptic curve $_$,  
+
of an elliptic curve $X$,  
$$_$$
+
$$j(X)=1728\frac{4a^3}{4a^3+27b^2}\in k$$
does not depend on the choice of the
+
does not depend on the choice of the
coordinate system. Two elliptic curves have the same $_$-invariant if
+
coordinate system. Two elliptic curves have the same $j$-invariant if
and only if they are biregularly isomorphic. For any $_$ there is an
+
and only if they are biregularly isomorphic. For any $j\in k$ there is an
elliptic curve $_$ over $_$ with $_$.
+
elliptic curve $X$ over $k$ with $j(X)=j$.
  
 
==The group structure on an elliptic curve.==
 
==The group structure on an elliptic curve.==
Let $_$ be a fixed point
+
Let $P_0\in X$ be a fixed point
on an elliptic curve $_$. The mapping $_$ assigning to a point $_$ the
+
on an elliptic curve $X$. The mapping $P\to P-P_0$ assigning to a point $P\in X$ the
[[Divisor|divisor]] $_$ on $_$ establishes a one-to-one correspondence
+
[[Divisor|divisor]] $P-P_0$ on $X$ establishes a one-to-one correspondence
between $_$ and the group $_$ of divisor classes of degree $_$ on $_$,
+
between $X$ and the group ${\rm Pic}^0\; X$ of divisor classes of degree $0$ on $X$,
 
that is, the
 
that is, the
[[Picard variety|Picard variety]] of $_$. This correspondence endows
+
[[Picard variety|Picard variety]] of $X$. This correspondence endows
$_$ with the structure of an Abelian group that is compatible with the
+
$X$ with the structure of an Abelian group that is compatible with the
structure of an algebraic variety and that turns $_$ into a
+
structure of an algebraic variety and that turns $X$ into a
one-dimensional Abelian variety $_$; here $_$ is the trivial element
+
one-dimensional Abelian variety $(X,P_0)$; here $P_0$ is the trivial element
 
of the group. This group structure has the following geometric
 
of the group. This group structure has the following geometric
description. Let $_$ be a smooth plane cubic curve. Then the sum of
+
description. Let $X\subset {\mathbb P}^2$ be a smooth plane cubic curve. Then the sum of
two points $_$ and $_$ is defined by the rule $_$, where $_$ is the
+
two points $P$ and $Q$ is defined by the rule $P+Q=P_0\circ (P\circ Q)$, where $P\circ Q$ is the
third point of intersection of $_$ with the line passing through $_$
+
third point of intersection of $X$ with the line passing through $P$
and $_$. In other words, the sum of three points on $_$ vanishes if
+
and $Q$. In other words, the sum of three points on $X$ vanishes if
 
and only if the points are collinear.
 
and only if the points are collinear.
  
 
==An elliptic curve as a one-dimensional Abelian variety.==
 
==An elliptic curve as a one-dimensional Abelian variety.==
Let $_$
+
Let $n_X$
denote the endomorphism of multiplication by $_$ in $_$. If $_$ is an
+
denote the endomorphism of multiplication by $n\in {\mathbb Z}$ in $(X,P_0)$. If $(Y,Q_0)$ is an
elliptic curve with distinguished point $_$, then any rational mapping
+
elliptic curve with distinguished point $Q_0$, then any rational mapping
$_$ has the form $_$, where $_$ and $_$ is a homomorphism of Abelian
+
$f:X\to Y$ has the form $f(P) = H(P) + Q_1$, where $Q_1 = f(P_0)\in Y$ and $h:(X,P_0) \to(Y,Q_0)$ is a homomorphism of Abelian
varieties. Here $_$ is either a constant mapping at $_$ or is an
+
varieties. Here $h$ is either a constant mapping at $Q_0$ or is an
 
[[Isogeny|isogeny]], that is, there is a homomorphism of Abelian
 
[[Isogeny|isogeny]], that is, there is a homomorphism of Abelian
varieties $_$ such that $_$ and $_$ for some $_$ (see
+
varieties $g:(Y,Q_0)\to(X,P_0)$ such that $gh=n_X$ and $hg=n_Y$  
[[#References|[1]]],
+
for some $n$ (see
[[#References|[6]]]).
+
{{Cite|Ca}},
 +
{{Cite|Ha}}).
  
The automorphism group of an elliptic curve $_$ acts transitively on
+
The automorphism group of an elliptic curve $X$ acts transitively on
$_$, and its subgroup $_$ of automorphisms leaving $_$ fixed is
+
$X$, and its subgroup $G={\rm Aut}(X,P_0)$ of automorphisms leaving $P_0$ fixed is
non-trivial and finite. Suppose that $_$ is not $_$ or $_$. When $_$
+
non-trivial and finite. Suppose that ${\rm char}\;k$ is not $2$ or $3$. When $j(X)$
is neither 0 nor $_$, then $_$ consists of the two elements $_$ and
+
is neither 0 nor $1728$, then $G$ consists of the two elements $1_X$ and
$_$. The order of $_$ is 4 when $_$ and 6 when $_$ (see
+
$(-1)_X$. The order of $G$ is 4 when $j(X)=1728$ and 6 when $j(X)=0$ (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[6]]],
+
{{Cite|Ha}},
[[#References|[13]]]).
+
{{Cite|Ta}}).
  
 
An important invariant of an elliptic curve is the endomorphism ring
 
An important invariant of an elliptic curve is the endomorphism ring
$_$ of the Abelian variety $_$. The mapping $_$ defines an imbedding
+
$ R={\rm End}(X,P_0) $ of the Abelian variety $(X,P_0)$.  
of $_$ in $_$. If $_$, one says that $_$ is an elliptic curve with
+
The mapping $n\mapsto n_X$ defines an imbedding
complex multiplication. The ring $_$ can be of one of the following
+
of ${\mathbb Z} $ in $R$. If $R\ne {\mathbb Z}$, one says that $X$ is an elliptic curve with
 +
complex multiplication. The ring $R$ can be of one of the following
 
types (see
 
types (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[9]]],
+
{{Cite|La2}},
[[#References|[13]]]): I) $_$; II) $_$, where $_$ is the ring of
+
{{Cite|Ta}}): i) $R={\mathbb Z}$; ii) $R={\mathbb Z}+f{\mathcal O}\subset {\mathcal O}$, where $\mathcal O$ is the ring of
algebraic integers of an imaginary quadratic field $_$ and $_$; or
+
algebraic integers of an imaginary quadratic field $k$ and $f\in {\mathbb N}$; or
III) $_$ is a non-commutative $_$-algebra of rank 4 without divisors
+
iii) $R$ is a non-commutative ${\mathbb Z}$-algebra of rank 4 without divisors
of zero. In this case $_$ and $_$ is a maximal order in the quaternion
+
of zero. In this case $p={\rm char}\; k > 0$ and $R$ is a maximal order in the quaternion
algebra over $_$ ramified only at $_$ and $_$. Such elliptic curves
+
algebra over ${\mathbb Q}$ ramified only at $p$ and $\infty$. Such elliptic curves
exist for all $_$ and are called supersingular; elliptic curves in
+
exist for all $p$ and are called supersingular; elliptic curves in
characteristic $_$ that are not supersingular are said to be ordinary.
+
characteristic $p$ that are not supersingular are said to be ordinary.
  
The group $_$ of points of an elliptic curve $_$ with orders that
+
The group $X_n = {\rm Ker}\; n_X$ of points of an elliptic curve $X$ with orders that
divide $_$ has the following structure: $_$ when $_$. For $_$ and
+
divide $n$ has the following structure:  
ordinary elliptic curves $_$, while for supersingular elliptic curves
+
$X_n\approx ({\mathbb Z}/n{\mathbb Z})^2$ when $(n,{\rm char}\; k)=1 $. For ${\rm char}\; k = p >0$ and
$_$. For a prime number $_$ the
+
ordinary elliptic curves $X_p\cong {\mathbb Z}/p{\mathbb Z}$, while for supersingular elliptic curves
[[Tate module|Tate module]] $_$ is isomorphic to $_$.
+
$X_p\cong \{0\}$. For a prime number $l\ne {\rm char}\; k$ the
 +
[[Tate module|Tate module]] $T_l(X)$ is isomorphic to ${\mathbb Z}_l^2$.
  
 
==Elliptic curves over non-closed fields.==
 
==Elliptic curves over non-closed fields.==
Let $_$ be an elliptic
+
Let $X$ be an elliptic
curve over an arbitrary field $_$. If the set of $_$-rational points
+
curve over an arbitrary field $k$. If the set of $k$-rational points
$_$ of $_$ is not empty, then $_$ is biregularly isomorphic to a plane
+
$X(k)$ of $X$ is not empty, then $X$ is biregularly isomorphic to a plane
cubic curve (1) with $_$ ($_$). The point at infinity $_$ of (1) is
+
cubic curve (1) with $a,b\in k$ (${\rm char}\; k \ne 2,3$). The point at infinity $P_0$ of (1) is
defined over $_$. As above, one can introduce a group structure on
+
defined over $k$. As above, one can introduce a group structure on
(1), turning $_$ into a one-dimensional Abelian variety over $_$ and
+
(1), turning $X$ into a one-dimensional Abelian variety over $k$ and
turning the set $_$ into an Abelian group with $_$ as trivial
+
turning the set $X(k)$ into an Abelian group with $P_0$ as trivial
element. If $_$ is finitely generated over its prime subfield, then
+
element. If $k$ is finitely generated over its prime subfield, then
$_$ is a finitely-generated group (the Mordell–Weil theorem).
+
$X(k)$ is a finitely-generated group (the Mordell–Weil theorem).
  
For any elliptic curve $_$ there is defined the
+
For any elliptic curve $X$ there is defined the
[[Jacobi variety|Jacobi variety]] $_$, which is a one-dimensional
+
[[Jacobi variety|Jacobi variety]] $J(X)$, which is a one-dimensional
Abelian variety over $_$, and $_$ is a
+
Abelian variety over $k$, and $X$ is a
 
[[Principal homogeneous space|principal homogeneous space]] over
 
[[Principal homogeneous space|principal homogeneous space]] over
$_$. If $_$ is not empty, then the choice of $_$ specifies an
+
$J(X)$. If $X(k)$ is not empty, then the choice of $P_0\in X(k)$ specifies an
isomorphism $_$ under which $_$ becomes the trivial element of $_$. In
+
isomorphism $X\simeq J(X)$ under which $P_0$ becomes the trivial element of $J(X)$. In
general, $_$ and $_$ are isomorphic over a finite extension of $_$
+
general, $X$ and $J(X)$ are isomorphic over a finite extension of $k$
 
(see
 
(see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[4]]],
+
{{Cite|CaFr}},
[[#References|[13]]]).
+
{{Cite|Ta}}).
  
 
==Elliptic curves over the field of complex numbers.==
 
==Elliptic curves over the field of complex numbers.==
 
An elliptic
 
An elliptic
curve over $_$ is a compact
+
curve over ${\mathbb C}$ is a compact
 
[[Riemann surface|Riemann surface]] of genus 1, and vice versa. The
 
[[Riemann surface|Riemann surface]] of genus 1, and vice versa. The
group structure turns $_$ into a complex Lie group, which is a
+
group structure turns $X$ into a complex Lie group, which is a
one-dimensional complex torus $_$, where $_$ is a lattice in the
+
one-dimensional complex torus ${\mathbb C}/\Lambda$, where $\Lambda$ is a lattice in the
complex plane $_$. Conversely, any one-dimensional complex torus is an
+
complex plane ${\mathbb C}$. Conversely, any one-dimensional complex torus is an
 
elliptic curve (see
 
elliptic curve (see
[[#References|[3]]]). From the topological point of view, an elliptic
+
{{Cite|Mu}}). From the topological point of view, an elliptic
 
curve is a two-dimensional torus.
 
curve is a two-dimensional torus.
  
The theory of elliptic curves over $_$ is in essence equivalent to the
+
The theory of elliptic curves over ${\mathbb C}$ is in essence equivalent to the
theory of elliptic functions. An identification of a torus $_$ with an
+
theory of elliptic functions. An identification of a torus ${\mathbb C}/\Lambda$ with an
 
elliptic curve can be effected as follows. The elliptic functions with
 
elliptic curve can be effected as follows. The elliptic functions with
a given period lattice $_$ form a field generated by the Weierstrass
+
a given period lattice $\Lambda$ form a field generated by the Weierstrass
$_$-function (see
+
$wp$-function (see
 
[[Weierstrass elliptic functions|Weierstrass elliptic functions]]) and
 
[[Weierstrass elliptic functions|Weierstrass elliptic functions]]) and
its derivative $_$, which are connected by the relation  
+
its derivative $\wp'(z)$, which are connected by the relation  
$$_$$
+
$$\wp'=4\wp^3 - g_2\wp - g_3$$
The
+
The
mapping $_$ ($_$) induces an isomorphism between the torus $_$ and the
+
mapping ${\mathbb C}\to {\mathbb P}^2$ (${z} \mapsto (1:\wp({z}):\wp'({z}))$) induces an isomorphism between the torus ${\mathbb C}/\Lambda$ and the
elliptic curve $_$ with equation $_$. The identification of $_$ given
+
elliptic curve $X\subset{\mathbb P^2}$ with equation $y^2=4x^3-g_2x-g_3$. The identification of $X$ given
by (1) with the torus $_$ is effected by curvilinear integrals of the
+
by (1) with the torus ${\mathbb C}/\Lambda$ is effected by curvilinear integrals of the
holomorphic form $_$ and gives an isomorphism $_$.
+
holomorphic form $\omega = dx/y$ and gives an isomorphism $X\simeq J(X)$.
  
The description of the set of all elliptic curves as tori $_$ leads to
+
The description of the set of all elliptic curves as tori ${\mathbb C}/\Lambda$ leads to
 
the
 
the
[[Modular function|modular function]] $_$. Two lattices $_$ and $_$
+
[[Modular function|modular function]] $J(\tau)$. Two lattices $\Lambda$ and $\Lambda'$
 
determine isomorphic tori if and only if they are similar, that is, if
 
determine isomorphic tori if and only if they are similar, that is, if
 
one is obtained from the other by multiplication by a complex
 
one is obtained from the other by multiplication by a complex
number. Therefore it may be assumed that $_$ is generated by the
+
number. Therefore it may be assumed that $\Lambda$ is generated by the
numbers 1 and $_$ in $_$. Two lattices with bases $_$ and $_$ are
+
numbers 1 and $\tau$ in $H=\{\tau \in {\mathbb C}: {\rm Im}\; \tau > 0$. Two lattices with bases $1,\tau$ and $1,\tau'$ are
similar if and only if $_$ for an element $_$ of the
+
similar if and only if $\tau'=\gamma(\tau)$ for an element $\gamma$ of the
[[Modular group|modular group]] $_$. The modular function  
+
[[Modular group|modular group]] $\Gamma$. The modular function  
$$_$$
+
$$J(\tau)=\frac{g_2^3}{g_2^3-27g_3^2}$$
is
+
is
also called the absolute invariant; $_$ if and only if $_$ for some
+
also called the absolute invariant; $J(\tau)=J(\tau')$ if and only if $\tau'=\gamma(\tau)$ for some
$_$, and the function $_$ produces a one-to-one correspondence between
+
$\gamma\in\Gamma$, and the function $J:H/\Gamma\to{\mathbb C}$ produces a one-to-one correspondence between
the classes of isomorphic elliptic curves over $_$ and the complex
+
the classes of isomorphic elliptic curves over ${\mathbb C}$ and the complex
numbers. If $_$, then $_$.
+
numbers. If $X={\mathbb C}/\Lambda$, then $j(X)=1728J(\tau)$.
  
An elliptic curve $_$ has complex multiplication if and only if $_$ is
+
An elliptic curve $X$ has complex multiplication if and only if $\tau$ is
 
an imaginary
 
an imaginary
[[Quadratic irrationality|quadratic irrationality]]. In this case $_$
+
[[Quadratic irrationality|quadratic irrationality]]. In this case ${\mathbb R}$
 
is a subring of finite index in the ring of algebraic integers of the
 
is a subring of finite index in the ring of algebraic integers of the
imaginary quadratic field $_$. Elliptic curves with complex
+
imaginary quadratic field ${\mathbb Q}(\tau)$. Elliptic curves with complex
 
multiplication are closely connected with the
 
multiplication are closely connected with the
 
[[Class field theory|class field theory]] for imaginary quadratic
 
[[Class field theory|class field theory]] for imaginary quadratic
 
fields (see
 
fields (see
[[#References|[4]]],
+
{{Cite|CaFr}},
[[#References|[8]]]).
+
{{Cite|La}}).
  
 
==Arithmetic of elliptic curves.==
 
==Arithmetic of elliptic curves.==
Let $_$ be an elliptic curve over
+
Let $X$ be an elliptic curve over
the finite field $_$ with $_$ elements. The set $_$ is always
+
the finite field $k$ with $q$ elements. The set $X(k)$ is always
non-empty and finite. Hence $_$ is endowed with the structure of a
+
non-empty and finite. Hence $X$ is endowed with the structure of a
one-dimensional Abelian variety over $_$, and $_$ with that of a
+
one-dimensional Abelian variety over $k$, and $X(k)$ with that of a
finite Abelian group. The order $_$ of $_$ satisfies $_$. The
+
finite Abelian group. The order $A$ of $X(k)$ satisfies $|q+1-A|\le 2 \sqrt{q}$. The
 
characteristic polynomial of the
 
characteristic polynomial of the
 
[[Frobenius endomorphism|Frobenius endomorphism]] acting on the Tate
 
[[Frobenius endomorphism|Frobenius endomorphism]] acting on the Tate
module $_$, $_$, is $_$. Its roots $_$ and $_$ are complex-conjugate
+
module $T_l(X)$, $l\ne {\rm char}\; k$, is $t^2-(q+1-A)t + q$. Its roots $\alpha$ and $\bar \alpha$ are complex-conjugate
algebraic integers of modulus $_$. For any finite extension $_$ of $_$
+
algebraic integers of modulus $\sqrt{q}$. For any finite extension $k_n$ of $k$
of degree $_$, the order of $_$ is $_$. The
+
of degree $n$, the order of $X(k_n)$ is $q^n+1-(\alpha^n+{\bar \alpha}^n)$. The
[[Zeta-function|zeta-function]] of $_$ is  
+
[[Zeta-function|zeta-function]] of $X$ is  
$$_$$
+
$$\frac{(1-q^{-s})(1-q^{1-s})}{1-(q+1-A)q^{-s}+q^{1-2s}}.$$
For any algebraic
+
For any algebraic
integer $_$ of modulus $_$ in some imaginary quadratic field (or in
+
integer $\alpha$ of modulus $\sqrt{q}$ in some imaginary quadratic field (or in
$_$) one can find an elliptic curve $_$ over $_$ such that the order
+
${\mathbb Q}$) one can find an elliptic curve $X$ over $k$ such that the order
of $_$ is $_$.
+
of $X(k)$ is $q+1-(\alpha+{\bar\alpha})$.
  
Let $_$ be the field $_$ of $_$-adic numbers or a finite algebraic
+
Let $k$ be the field ${\mathbb Q}_p$ of $p$-adic numbers or a finite algebraic
extension of it, let $_$ be the ring of integers of $_$, let $_$ be an
+
extension of it, let $B$ be the ring of integers of $k$, let $X$ be an
elliptic curve over $_$, and suppose that $_$ is non-empty. The group
+
elliptic curve over $k$, and suppose that $X(k)$ is non-empty. The group
structure turns $_$ into a commutative compact one-dimensional
+
structure turns $X(k)$ into a commutative compact one-dimensional
$_$-adic Lie group (cf.
+
$p$-adic Lie group (cf.
[[Lie-group-adic|Lie group, $_$-adic]]). The group $_$ is
+
[[Lie-group, p-adic|Lie group, $p$-adic]]). The group $X(k)$ is
 
Pontryagin-dual to the
 
Pontryagin-dual to the
[[Weil–Châtelet group|Weil–Châtelet group]] $_$. If $_$, then $_$ is a
+
[[Weil–Châtelet group|Weil–Châtelet group]] ${\rm WC}(k,X)$. If $j(X)\notin B$, then $X$ is a
 
Tate curve (see
 
Tate curve (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[5]]]) and there exists a canonical uniformization of
+
{{Cite|Ma}}) and there exists a canonical uniformization of
$_$ analogous to the case of $_$.
+
$X(k)$ analogous to the case of ${\mathbb C}$.
  
Let $_$ be an elliptic curve over $_$ for which $_$ is not empty. Then
+
Let $X$ be an elliptic curve over ${\mathbb Q}$ for which $X({\mathbb Q})$ is not empty. Then
$_$ is biregularly isomorphic to the curve (1) with $_$. Of all curves
+
$X$ is biregularly isomorphic to the curve (1) with $a,b\in {\mathbb Z}$. Of all curves
of the form (1) that are isomorphic to $_$ with integers $_$ and $_$,
+
of the form (1) that are isomorphic to $X$ with integers $a$ and $b$,
 
one chooses the one for which the absolute value of the discriminant
 
one chooses the one for which the absolute value of the discriminant
$_$ is minimal. The conductor $_$ and the $_$-function $_$ of $_$ are
+
$\Delta$ is minimal. The conductor $N$ and the $L$-function $L(X,s)$ of $X$ are
 
defined as formal products of local factors:  
 
defined as formal products of local factors:  
$$_$$
+
$$N=\prod f_p,\qquad L(X,s) = \prod L_p(X,s)$$
over all prime
+
over all prime
numbers $_$ (see
+
numbers $p$ (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[5]]],
+
{{Cite|Ma}},
[[#References|[13]]]). Here $_$ is some power of $_$, and $_$ is a
+
{{Cite|Ta}}). Here $f_p$ is some power of $p$, and
meromorphic function of the complex variable $_$ that has neither a
+
$L_p(X,s)$ is a
zero nor a pole at $_$. To determine the local factors one considers
+
meromorphic function of the complex variable $s$ that has neither a
the reduction of $_$ modulo $_$ ($_$), which is a plane projective
+
zero nor a pole at $s=1$. To determine the local factors one considers
curve $_$ over the residue class field $_$ and is given in an affine
+
the reduction of $X$ modulo $p$ ($2,\;3$), which is a plane projective
 +
curve $X_p$ over the residue class field ${\mathbb Z}/(p)$ and is given in an affine
 
coordinate system by the equation  
 
coordinate system by the equation  
$$_$$
+
$$y^2=x^3+{\bar\alpha}x+{\bar\beta}\qquad ({\bar\alpha}\equiv\alpha \;{\rm mod}\; p,\; {\bar\beta}\equiv\beta\;{\rm mod}\; p).$$
Let $_$ be the number of
+
Let $A_p$ be the number of
$_$-points on $_$. If $_$ does not divide $_$, then $_$ is an elliptic
+
${\mathbb Z}/(p)$-points on $X_p$. If $p$ does not divide $\Delta$, then $X_p$ is an elliptic
curve over $_$, and one puts  
+
curve over ${\mathbb Z}/(p)$, and one puts  
$$_$$
+
$$f_p=1,\qquad L_p(X,s) = \frac{1}{1-(p+1-A_p)p^{-s}+p{1-2s}}.$$
If $_$ divides $_$, then the
+
If $p$ divides $\Delta$, then the
polynomial $_$ has a multiple root, and one puts  
+
polynomial $x^2+{\bar a}+{\bar b}$ has a multiple root, and one puts  
$$_$$
+
$$L_p(X,s) = \frac{1}{1-(p+1-A_p)p^{-s}},\qquad f_p=p^2 \text{ or } p$$
(depending on
+
(depending on
 
whether it is a triple or a double root). The product (2) converges in
 
whether it is a triple or a double root). The product (2) converges in
the right half-plane $_$. It has been conjectured that $_$ has a
+
the right half-plane ${\rm Re}\; s > 3/2$. It has been conjectured that $L(X,s)$ has a
 
meromorphic extension to the whole complex plane and that the function
 
meromorphic extension to the whole complex plane and that the function
  
$$_$$
+
$$\xi_X(s) = N^{s/2}(2\pi)^{-s}\;\Gamma(s)L(X,s)$$
(where $_$ is the
+
(where $\Gamma(s)$ is the
 
[[Gamma-function|gamma-function]]) satisfies the functional equation
 
[[Gamma-function|gamma-function]]) satisfies the functional equation
$_$ with $_$ (see
+
$\xi_X(s) = W\xi_X(2-s)$ with $W=\pm1$ (see
[[#References|[5]]],
+
{{Cite|Ma}},
[[#References|[3]]]). This conjecture has been proved for elliptic
+
{{Cite|Mu}}). This conjecture has been proved for elliptic
 
curves with complex multiplication.
 
curves with complex multiplication.
  
The group $_$ is isomorphic to $_$, where $_$ is a finite Abelian
+
The group $X({\mathbb Q})$ is isomorphic to $F\oplus X({\mathbb Q})_t$, where $X({\mathbb Q})_t$ is a finite Abelian
group and $_$ is a free Abelian group of a certain finite rank
+
group and $F$ is a free Abelian group of a certain finite rank
$_$. $_$ is isomorphic to one of the following 15 groups (see
+
$r$. $X({\mathbb Q})_t$ is isomorphic to one of the following 15 groups (see
[[#References|[11]]]): $_$, $_$ or $_$, and $_$, $_$. The number $_$
+
{{Cite|SeDeKu}}): ${\mathbb Z}/m{\mathbb Z}$, $1\le m\le 10$ or $m = 12$, and $({\mathbb Z}/2{\mathbb Z})\times ({\mathbb Z}/\nu{\mathbb Z})$, $1\le \nu\le 4$. The number $r$
is called the rank of the elliptic curve over $_$, or its
+
is called the rank of the elliptic curve over ${\mathbb Q}$, or its
$_$-rank. Examples are known of elliptic curves over $_$ of rank
+
${\mathbb Q}$-rank. Examples are known of elliptic curves over ${\mathbb Q}$ of rank
$_$. There is a conjecture (see
+
$\ge 12$. There is a conjecture (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[13]]]) that over $_$ there exist elliptic curves of
+
{{Cite|Ta}}) that over ${\mathbb Q}$ there exist elliptic curves of
 
arbitrary large rank.
 
arbitrary large rank.
  
In the study of $_$ one uses the Tate height $_$, which is a
+
In the study of $X({\mathbb Q})$ one uses the Tate height ${\hat h}:X({\mathbb Q})\to {\mathbb R}^+$, which is a
non-negative definite quadratic form on $_$ (see
+
non-negative definite quadratic form on $X({\mathbb Q})$ (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[3]]],
+
{{Cite|Mu}},
[[#References|[8]]], and also
+
{{Cite|La}}, and also
 
[[Height, in Diophantine geometry|Height, in Diophantine
 
[[Height, in Diophantine geometry|Height, in Diophantine
geometry]]). For any $_$ the set $_$ is finite. In particular, $_$
+
geometry]]). For any $c\in {\mathbb R}^+$ the set $\{P\in X(\mathbb Q) | {\hat h}(P)\le c\}$ is finite. In particular, $\hat h$
vanishes precisely on the torsion subgroup of $_$.
+
vanishes precisely on the torsion subgroup of $X({\mathbb Q})_t$.
  
 
An important invariant of an elliptic curve is its Tate–Shafarevich
 
An important invariant of an elliptic curve is its Tate–Shafarevich
group $_$ (see
+
group ${\rm Sha}\;(X)$ (see
 
[[Weil–Châtelet group|Weil–Châtelet group]]). The non-trivial elements
 
[[Weil–Châtelet group|Weil–Châtelet group]]). The non-trivial elements
of $_$, an elliptic curve without $_$-points, provide examples of
+
of ${\rm Sha}\;(X)$, an elliptic curve without ${\mathbb Q}$-points, provide examples of
 
elliptic curves for which the
 
elliptic curves for which the
[[Hasse principle|Hasse principle]] fails to hold. The group $_$ is
+
[[Hasse principle|Hasse principle]] fails to hold. The group ${\rm Sha}\;(X)$ is
periodic and for every $_$ the subgroup of its elements of order
+
periodic and for every $n$ the subgroup of its elements of order
dividing $_$ is finite. For a large number of elliptic curves it has
+
dividing $n$ is finite. For a large number of elliptic curves it has
been verified that the 2- and $_$-components of $_$ are finite (see
+
been verified that the 2- and $3$-components of ${\rm Sha}$ are finite (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[4]]],
+
{{Cite|CaFr}},
[[#References|[5]]]). There is a conjecture that $_$ is finite.
+
{{Cite|Ma}}). There is a conjecture that ${\rm Sha}$ is finite.
  
 
A conjecture of Birch and Swinnerton-Dyer asserts (see
 
A conjecture of Birch and Swinnerton-Dyer asserts (see
[[#References|[5]]],
+
{{Cite|Ma}},
[[#References|[13]]]) that the order of the zero of the $_$-function
+
{{Cite|Ta}}) that the order of the zero of the $L$-function
$_$ at $_$ is equal to the $_$-rank of $_$. In particular, $_$ has a
+
$L(X,s)$ at $s=1$ is equal to the ${\mathbb Q}$-rank of $X$. In particular, $L(X,s)$ has a
zero at $_$ if and only if $_$ is infinite. So far (1984) the
+
zero at $s=1$ if and only if $X({\mathbb Q})$ is infinite. So far (1984) the
 
conjecture has not been proved for a single elliptic curve, but for
 
conjecture has not been proved for a single elliptic curve, but for
elliptic curves with complex multiplication (and $_$) it has been
+
elliptic curves with complex multiplication (and $j=1$) it has been
established that when $_$ is infinite, then the $_$-function has a
+
established that when $X({\mathbb Q})$ is infinite, then the $L$-function has a
zero at $_$ (see
+
zero at $s=1$ (see
[[#References|[14]]]). The conjecture of Birch and Swinnerton-Dyer
+
{{Cite|CoWi}}). The conjecture of Birch and Swinnerton-Dyer
 
gives the principal term of the asymptotic expansion of the
 
gives the principal term of the asymptotic expansion of the
$_$-function as $_$; in it there occur the orders of the groups $_$
+
$L$-function as $s\to 1$; in it there occur the orders of the groups ${\rm Sha}\;(X)$
and $_$ and the determinant of the Tate height
+
and $X({\mathbb Q})_t$ and the determinant of the Tate height
[[#References|[1]]]. It can be restated in terms of the Tamagawa
+
{{Cite|Ca}}. It can be restated in terms of the Tamagawa
 
numbers (cf.
 
numbers (cf.
 
[[Tamagawa number|Tamagawa number]], see
 
[[Tamagawa number|Tamagawa number]], see
[[#References|[7]]]).
+
{{Cite|Bl}}).
  
There is a conjecture of Weil that an elliptic curve $_$ has a
+
There is a conjecture of Weil that an elliptic curve $X$ has a
 
uniformization by modular functions relative to the congruence
 
uniformization by modular functions relative to the congruence
subgroup $_$ of the modular group $_$ (see
+
subgroup $\Gamma_0(N)$ of the modular group $\Gamma$ (see
[[#References|[5]]] and also
+
{{Cite|Ma}} and also
 
[[Zeta-function|Zeta-function]] in algebraic geometry). This
 
[[Zeta-function|Zeta-function]] in algebraic geometry). This
 
conjecture has been proved for elliptic functions with complex
 
conjecture has been proved for elliptic functions with complex
 
multiplication. It is known (see
 
multiplication. It is known (see
[[#References|[15]]]) that every algebraic curve over $_$ can be
+
{{Cite|Be}}) that every algebraic curve over $\mathbb Q$ can be
 
uniformized (cf.
 
uniformized (cf.
 
[[Uniformization|Uniformization]]) by modular functions relative to
 
[[Uniformization|Uniformization]]) by modular functions relative to
some subgroup of $_$ of finite index.
+
some subgroup of $\Gamma$ of finite index.
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD
 
valign="top"> J.W.S. Cassels, "Diophantine equations with special
 
reference to elliptic curves" ''J. London Math. Soc.'' , '''41'''
 
(1966) pp. 193–291</TD></TR><TR><TD valign="top">[2]</TD> <TD
 
valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine
 
Funktionentheorie und elliptische Funktionen" , Springer
 
(1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
 
D. Mumford, "Abelian varieties" , Oxford Univ. Press
 
(1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
 
J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory''
 
, Acad. Press (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD
 
valign="top"> Yu.I. Manin, "Cyclotomic fields and modular curves"
 
''Russian Math. Surveys'' , '''26''' : 6 (1971) pp. 6–78 ''Uspekhi
 
Mat. Nauk'' , '''26''' : 6 (1971) pp. 7–71</TD></TR><TR><TD
 
valign="top">[6]</TD> <TD valign="top"> R. Hartshorne, "Algebraic
 
geometry" , Springer (1977) pp. 91</TD></TR><TR><TD
 
valign="top">[7]</TD> <TD valign="top"> S. Bloch, "A note on height
 
pairings, Tamagawa numbers, and the Birch and Swinnnerton-Dyer
 
conjecture" ''Invent. Math.'' , '''58''' (1980)
 
pp. 65–76</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">
 
S. Lang, "Elliptic curves; Diophantine analysis" , Springer
 
(1978)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">
 
S. Lang, "Elliptic functions" , Addison-Wesley (1973)</TD></TR><TR><TD
 
valign="top">[10]</TD> <TD valign="top"> B. Mazur, "Rational isogenies
 
of prime degree" ''Invent. Math.'' , '''44''' (1978)
 
pp. 129–162</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">
 
J.-P. Serre (ed.)  P. Deligne (ed.)  W. Kuyk (ed.) , ''Modular
 
functions of one variable. 4'' , ''Lect. notes in math.'' , '''476'''
 
, Springer (1975)</TD></TR><TR><TD valign="top">[12]</TD> <TD
 
valign="top"> J.F. Mestre, "Construction d'une courbe elliptique de
 
rang $_$" ''C.R. Acad. Sci. Paris Sér. 1'' , '''295''' (1982)
 
pp. 643–644</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">
 
J. Tate, "The arithmetic of elliptic curves" ''Invent. Math.'' ,
 
'''23''' (1974) pp. 197–206</TD></TR><TR><TD valign="top">[14]</TD>
 
<TD valign="top"> J. Coates, A. Wiles, "On the conjecture of Birch and
 
Swinnerton-Dyer" ''Invent. Math.'' , '''39''' (1977)
 
pp. 223–251</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">
 
G.V. Belyi, "On Galois extensions of a maximal cyclotomic field"
 
''Math. USSR Izv.'' , '''14''' : 2 (1980) pp. 247–256
 
''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''43''' (1979)
 
pp. 267–276</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
 
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD
+
{|
valign="top"> B. Mazur, "Modular curves and the Eisenstein ideal"
+
|-
''Publ. Math. IHES'' , '''47''' (1978) pp. 33–186</TD></TR><TR><TD
+
|valign="top"|{{Ref|Be}}||valign="top"| G.V. Belyi, "On Galois extensions of a maximal cyclotomic field" ''Math. USSR Izv.'', '''14''' : 2 (1980) pp. 247–256 ''Izv. Akad. Nauk SSSR Ser. Mat.'', '''43''' (1979) pp. 267–276  {{ZBL|0429.12004}}         
valign="top">[a2]</TD> <TD valign="top"> J.H. Silverman, "The
+
|-
arithmetic of elliptic curves" , Springer (1986)</TD></TR></table>
+
|valign="top"|{{Ref|Bl}}||valign="top"| S. Bloch, "A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture" ''Invent. Math.'', '''58''' (1980) pp. 65–76    {{MR|0570874}}  {{ZBL|0444.14015}}     
 +
|-
 +
|valign="top"|{{Ref|Ca}}||valign="top"| J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" ''J. London Math. Soc.'', '''41''' (1966) pp. 193–291  {{MR|0199150}}         
 +
|-
 +
|valign="top"|{{Ref|CaFr}}||valign="top"| J.W.S. Cassels (ed.)  A. Fröhlich (ed.), ''Algebraic number theory'', Acad. Press (1967) {{MR|0215665}}  {{ZBL|0153.07403}}       
 +
|-
 +
|valign="top"|{{Ref|CoWi}}||valign="top"| J. Coates, A. Wiles, "On the conjecture of Birch and Swinnerton-Dyer" ''Invent. Math.'', '''39''' (1977) pp. 223–251  {{MR|0463176}}  {{ZBL|0359.14009}}         
 +
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"| R. Hartshorne, "Algebraic geometry", Springer (1977) pp. 91  {{MR|0463157}}  {{ZBL|0367.14001}}         
 +
|-
 +
|valign="top"|{{Ref|HuCo}}||valign="top"| A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen", Springer (1964)  {{MR|0173749}}  {{ZBL|0135.12101}}
 +
|-
 +
|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Elliptic curves: Diophantine analysis", Springer (1978)    {{MR|0518817}}  {{ZBL|0388.10001}}   
 +
|-
 +
|valign="top"|{{Ref|La2}}||valign="top"| S. Lang, "Elliptic functions", Addison-Wesley (1973)  {{MR|0409362}}  {{ZBL|0316.14001}}         
 +
|-
 +
|valign="top"|{{Ref|Ma}}||valign="top"| Yu.I. Manin, "Cyclotomic fields and modular curves" ''Russian Math. Surveys'', '''26''' : 6 (1971) pp. 6–78 ''Uspekhi Mat. Nauk'', '''26''' : 6 (1971) pp. 7–71  {{MR|0401653}}         
 +
|-
 +
|valign="top"|{{Ref|Ma2}}||valign="top"| B. Mazur, "Rational isogenies of prime degree" ''Invent. Math.'', '''44''' (1978) pp. 129–162  {{MR|0482230}}  {{ZBL|0386.14009}}         
 +
|-
 +
|valign="top"|{{Ref|Ma3}}||valign="top"| B. Mazur, "Modular curves and the Eisenstein ideal" ''Publ. Math. IHES'', '''47''' (1977) pp. 33–186           {{MR|0488287}}  {{ZBL|0394.14008}}
 +
|-
 +
|valign="top"|{{Ref|Me}}||valign="top"| J.F. Mestre, "Construction d'une courbe elliptique de rang $\ge 12$" ''C.R. Acad. Sci. Paris Sér. 1'', '''295''' (1982) pp. 643–644  {{MR|0688896}}  {{ZBL|0541.14027}}         
 +
|-
 +
|valign="top"|{{Ref|Mu}}||valign="top"| D. Mumford, "Abelian varieties", Oxford Univ. Press (1974)  {{ZBL|0326.14012}}         
 +
|-
 +
|valign="top"|{{Ref|SeDeKu}}||valign="top"| J.-P. Serre (ed.)  P. Deligne (ed.)  W. Kuyk (ed.), ''Modular functions of one variable. 4'', ''Lect. notes in math.'', '''476''', Springer (1975)  {{MR|0404145}} {{MR|0404146}}         
 +
|-
 +
|valign="top"|{{Ref|Si}}||valign="top"| J.H. Silverman, "The arithmetic of elliptic curves", Springer (1986) {{MR|0817210}}  {{ZBL|0585.14026}}         
 +
|-
 +
|valign="top"|{{Ref|Ta}}||valign="top"| J. Tate, "The arithmetic of elliptic curves" ''Invent. Math.'', '''23''' (1974) pp. 197–206  {{MR|0419359}}  {{ZBL|0296.14018}}         
 +
|-
 +
|}

Latest revision as of 20:37, 19 September 2017

2020 Mathematics Subject Classification: Primary: 14h57 Secondary: 11Gxx14K15 [MSN][ZBL]


An elliptic curve is a non-singular complete algebraic curve of genus 1. The theory of elliptic curves is the source of a large part of contemporary algebraic geometry. But historically the theory of elliptic curves arose as a part of analysis, as the theory of elliptic integrals and elliptic functions (cf. Elliptic integral; Elliptic function).

Examples. A non-singular plane projective cubic curve; the intersection of two non-singular quadrics in three-dimensional projective space; a two-sheeted covering of the projective line ramified at exactly four points; and also a one-dimensional Abelian variety are elliptic curves.

The geometry of an elliptic curve.

Let $X$ be an elliptic curve over an algebraically closed field $k$. Then $X$ is biregularly isomorphic to a plane cubic curve (see [Ca], [La2], [Ta]). If $\mathop{\mathrm{char}} k \ne 2,3$, then in the projective plane ${\mathbb P}^2$ there is an affine coordinate system in which the equation of $X$ is in normal Weierstrass form: $$y^2=x^3+ax+b$$ The curve $X$ is non-singular if and only if the polynomial $x^3+ax+b$ does not have multiple zeros, that is, if the discriminant $\Delta = -16(4a^3+27b^2)\ne 0$. In ${\mathbb P}^2$ the curve (1) has a unique point at infinity, which is denoted by $P_0$; $P_0$ is a point of inflection of (1), and the tangent at $P_0$ is the line at infinity. The $j$-invariant of an elliptic curve $X$, $$j(X)=1728\frac{4a^3}{4a^3+27b^2}\in k$$ does not depend on the choice of the coordinate system. Two elliptic curves have the same $j$-invariant if and only if they are biregularly isomorphic. For any $j\in k$ there is an elliptic curve $X$ over $k$ with $j(X)=j$.

The group structure on an elliptic curve.

Let $P_0\in X$ be a fixed point on an elliptic curve $X$. The mapping $P\to P-P_0$ assigning to a point $P\in X$ the divisor $P-P_0$ on $X$ establishes a one-to-one correspondence between $X$ and the group ${\rm Pic}^0\; X$ of divisor classes of degree $0$ on $X$, that is, the Picard variety of $X$. This correspondence endows $X$ with the structure of an Abelian group that is compatible with the structure of an algebraic variety and that turns $X$ into a one-dimensional Abelian variety $(X,P_0)$; here $P_0$ is the trivial element of the group. This group structure has the following geometric description. Let $X\subset {\mathbb P}^2$ be a smooth plane cubic curve. Then the sum of two points $P$ and $Q$ is defined by the rule $P+Q=P_0\circ (P\circ Q)$, where $P\circ Q$ is the third point of intersection of $X$ with the line passing through $P$ and $Q$. In other words, the sum of three points on $X$ vanishes if and only if the points are collinear.

An elliptic curve as a one-dimensional Abelian variety.

Let $n_X$ denote the endomorphism of multiplication by $n\in {\mathbb Z}$ in $(X,P_0)$. If $(Y,Q_0)$ is an elliptic curve with distinguished point $Q_0$, then any rational mapping $f:X\to Y$ has the form $f(P) = H(P) + Q_1$, where $Q_1 = f(P_0)\in Y$ and $h:(X,P_0) \to(Y,Q_0)$ is a homomorphism of Abelian varieties. Here $h$ is either a constant mapping at $Q_0$ or is an isogeny, that is, there is a homomorphism of Abelian varieties $g:(Y,Q_0)\to(X,P_0)$ such that $gh=n_X$ and $hg=n_Y$ for some $n$ (see [Ca], [Ha]).

The automorphism group of an elliptic curve $X$ acts transitively on $X$, and its subgroup $G={\rm Aut}(X,P_0)$ of automorphisms leaving $P_0$ fixed is non-trivial and finite. Suppose that ${\rm char}\;k$ is not $2$ or $3$. When $j(X)$ is neither 0 nor $1728$, then $G$ consists of the two elements $1_X$ and $(-1)_X$. The order of $G$ is 4 when $j(X)=1728$ and 6 when $j(X)=0$ (see [Ca], [Ha], [Ta]).

An important invariant of an elliptic curve is the endomorphism ring $ R={\rm End}(X,P_0) $ of the Abelian variety $(X,P_0)$. The mapping $n\mapsto n_X$ defines an imbedding of ${\mathbb Z} $ in $R$. If $R\ne {\mathbb Z}$, one says that $X$ is an elliptic curve with complex multiplication. The ring $R$ can be of one of the following types (see [Ca], [La2], [Ta]): i) $R={\mathbb Z}$; ii) $R={\mathbb Z}+f{\mathcal O}\subset {\mathcal O}$, where $\mathcal O$ is the ring of algebraic integers of an imaginary quadratic field $k$ and $f\in {\mathbb N}$; or iii) $R$ is a non-commutative ${\mathbb Z}$-algebra of rank 4 without divisors of zero. In this case $p={\rm char}\; k > 0$ and $R$ is a maximal order in the quaternion algebra over ${\mathbb Q}$ ramified only at $p$ and $\infty$. Such elliptic curves exist for all $p$ and are called supersingular; elliptic curves in characteristic $p$ that are not supersingular are said to be ordinary.

The group $X_n = {\rm Ker}\; n_X$ of points of an elliptic curve $X$ with orders that divide $n$ has the following structure: $X_n\approx ({\mathbb Z}/n{\mathbb Z})^2$ when $(n,{\rm char}\; k)=1 $. For ${\rm char}\; k = p >0$ and ordinary elliptic curves $X_p\cong {\mathbb Z}/p{\mathbb Z}$, while for supersingular elliptic curves $X_p\cong \{0\}$. For a prime number $l\ne {\rm char}\; k$ the Tate module $T_l(X)$ is isomorphic to ${\mathbb Z}_l^2$.

Elliptic curves over non-closed fields.

Let $X$ be an elliptic curve over an arbitrary field $k$. If the set of $k$-rational points $X(k)$ of $X$ is not empty, then $X$ is biregularly isomorphic to a plane cubic curve (1) with $a,b\in k$ (${\rm char}\; k \ne 2,3$). The point at infinity $P_0$ of (1) is defined over $k$. As above, one can introduce a group structure on (1), turning $X$ into a one-dimensional Abelian variety over $k$ and turning the set $X(k)$ into an Abelian group with $P_0$ as trivial element. If $k$ is finitely generated over its prime subfield, then $X(k)$ is a finitely-generated group (the Mordell–Weil theorem).

For any elliptic curve $X$ there is defined the Jacobi variety $J(X)$, which is a one-dimensional Abelian variety over $k$, and $X$ is a principal homogeneous space over $J(X)$. If $X(k)$ is not empty, then the choice of $P_0\in X(k)$ specifies an isomorphism $X\simeq J(X)$ under which $P_0$ becomes the trivial element of $J(X)$. In general, $X$ and $J(X)$ are isomorphic over a finite extension of $k$ (see [Ca], [CaFr], [Ta]).

Elliptic curves over the field of complex numbers.

An elliptic curve over ${\mathbb C}$ is a compact Riemann surface of genus 1, and vice versa. The group structure turns $X$ into a complex Lie group, which is a one-dimensional complex torus ${\mathbb C}/\Lambda$, where $\Lambda$ is a lattice in the complex plane ${\mathbb C}$. Conversely, any one-dimensional complex torus is an elliptic curve (see [Mu]). From the topological point of view, an elliptic curve is a two-dimensional torus.

The theory of elliptic curves over ${\mathbb C}$ is in essence equivalent to the theory of elliptic functions. An identification of a torus ${\mathbb C}/\Lambda$ with an elliptic curve can be effected as follows. The elliptic functions with a given period lattice $\Lambda$ form a field generated by the Weierstrass $wp$-function (see Weierstrass elliptic functions) and its derivative $\wp'(z)$, which are connected by the relation $$\wp'=4\wp^3 - g_2\wp - g_3$$ The mapping ${\mathbb C}\to {\mathbb P}^2$ (${z} \mapsto (1:\wp({z}):\wp'({z}))$) induces an isomorphism between the torus ${\mathbb C}/\Lambda$ and the elliptic curve $X\subset{\mathbb P^2}$ with equation $y^2=4x^3-g_2x-g_3$. The identification of $X$ given by (1) with the torus ${\mathbb C}/\Lambda$ is effected by curvilinear integrals of the holomorphic form $\omega = dx/y$ and gives an isomorphism $X\simeq J(X)$.

The description of the set of all elliptic curves as tori ${\mathbb C}/\Lambda$ leads to the modular function $J(\tau)$. Two lattices $\Lambda$ and $\Lambda'$ determine isomorphic tori if and only if they are similar, that is, if one is obtained from the other by multiplication by a complex number. Therefore it may be assumed that $\Lambda$ is generated by the numbers 1 and $\tau$ in $H=\{\tau \in {\mathbb C}: {\rm Im}\; \tau > 0$. Two lattices with bases $1,\tau$ and $1,\tau'$ are similar if and only if $\tau'=\gamma(\tau)$ for an element $\gamma$ of the modular group $\Gamma$. The modular function $$J(\tau)=\frac{g_2^3}{g_2^3-27g_3^2}$$ is also called the absolute invariant; $J(\tau)=J(\tau')$ if and only if $\tau'=\gamma(\tau)$ for some $\gamma\in\Gamma$, and the function $J:H/\Gamma\to{\mathbb C}$ produces a one-to-one correspondence between the classes of isomorphic elliptic curves over ${\mathbb C}$ and the complex numbers. If $X={\mathbb C}/\Lambda$, then $j(X)=1728J(\tau)$.

An elliptic curve $X$ has complex multiplication if and only if $\tau$ is an imaginary quadratic irrationality. In this case ${\mathbb R}$ is a subring of finite index in the ring of algebraic integers of the imaginary quadratic field ${\mathbb Q}(\tau)$. Elliptic curves with complex multiplication are closely connected with the class field theory for imaginary quadratic fields (see [CaFr], [La]).

Arithmetic of elliptic curves.

Let $X$ be an elliptic curve over the finite field $k$ with $q$ elements. The set $X(k)$ is always non-empty and finite. Hence $X$ is endowed with the structure of a one-dimensional Abelian variety over $k$, and $X(k)$ with that of a finite Abelian group. The order $A$ of $X(k)$ satisfies $|q+1-A|\le 2 \sqrt{q}$. The characteristic polynomial of the Frobenius endomorphism acting on the Tate module $T_l(X)$, $l\ne {\rm char}\; k$, is $t^2-(q+1-A)t + q$. Its roots $\alpha$ and $\bar \alpha$ are complex-conjugate algebraic integers of modulus $\sqrt{q}$. For any finite extension $k_n$ of $k$ of degree $n$, the order of $X(k_n)$ is $q^n+1-(\alpha^n+{\bar \alpha}^n)$. The zeta-function of $X$ is $$\frac{(1-q^{-s})(1-q^{1-s})}{1-(q+1-A)q^{-s}+q^{1-2s}}.$$ For any algebraic integer $\alpha$ of modulus $\sqrt{q}$ in some imaginary quadratic field (or in ${\mathbb Q}$) one can find an elliptic curve $X$ over $k$ such that the order of $X(k)$ is $q+1-(\alpha+{\bar\alpha})$.

Let $k$ be the field ${\mathbb Q}_p$ of $p$-adic numbers or a finite algebraic extension of it, let $B$ be the ring of integers of $k$, let $X$ be an elliptic curve over $k$, and suppose that $X(k)$ is non-empty. The group structure turns $X(k)$ into a commutative compact one-dimensional $p$-adic Lie group (cf. Lie group, $p$-adic). The group $X(k)$ is Pontryagin-dual to the Weil–Châtelet group ${\rm WC}(k,X)$. If $j(X)\notin B$, then $X$ is a Tate curve (see [Ca], [Ma]) and there exists a canonical uniformization of $X(k)$ analogous to the case of ${\mathbb C}$.

Let $X$ be an elliptic curve over ${\mathbb Q}$ for which $X({\mathbb Q})$ is not empty. Then $X$ is biregularly isomorphic to the curve (1) with $a,b\in {\mathbb Z}$. Of all curves of the form (1) that are isomorphic to $X$ with integers $a$ and $b$, one chooses the one for which the absolute value of the discriminant $\Delta$ is minimal. The conductor $N$ and the $L$-function $L(X,s)$ of $X$ are defined as formal products of local factors: $$N=\prod f_p,\qquad L(X,s) = \prod L_p(X,s)$$ over all prime numbers $p$ (see [Ca], [Ma], [Ta]). Here $f_p$ is some power of $p$, and $L_p(X,s)$ is a meromorphic function of the complex variable $s$ that has neither a zero nor a pole at $s=1$. To determine the local factors one considers the reduction of $X$ modulo $p$ ($2,\;3$), which is a plane projective curve $X_p$ over the residue class field ${\mathbb Z}/(p)$ and is given in an affine coordinate system by the equation $$y^2=x^3+{\bar\alpha}x+{\bar\beta}\qquad ({\bar\alpha}\equiv\alpha \;{\rm mod}\; p,\; {\bar\beta}\equiv\beta\;{\rm mod}\; p).$$ Let $A_p$ be the number of ${\mathbb Z}/(p)$-points on $X_p$. If $p$ does not divide $\Delta$, then $X_p$ is an elliptic curve over ${\mathbb Z}/(p)$, and one puts $$f_p=1,\qquad L_p(X,s) = \frac{1}{1-(p+1-A_p)p^{-s}+p{1-2s}}.$$ If $p$ divides $\Delta$, then the polynomial $x^2+{\bar a}+{\bar b}$ has a multiple root, and one puts $$L_p(X,s) = \frac{1}{1-(p+1-A_p)p^{-s}},\qquad f_p=p^2 \text{ or } p$$ (depending on whether it is a triple or a double root). The product (2) converges in the right half-plane ${\rm Re}\; s > 3/2$. It has been conjectured that $L(X,s)$ has a meromorphic extension to the whole complex plane and that the function

$$\xi_X(s) = N^{s/2}(2\pi)^{-s}\;\Gamma(s)L(X,s)$$ (where $\Gamma(s)$ is the gamma-function) satisfies the functional equation $\xi_X(s) = W\xi_X(2-s)$ with $W=\pm1$ (see [Ma], [Mu]). This conjecture has been proved for elliptic curves with complex multiplication.

The group $X({\mathbb Q})$ is isomorphic to $F\oplus X({\mathbb Q})_t$, where $X({\mathbb Q})_t$ is a finite Abelian group and $F$ is a free Abelian group of a certain finite rank $r$. $X({\mathbb Q})_t$ is isomorphic to one of the following 15 groups (see [SeDeKu]): ${\mathbb Z}/m{\mathbb Z}$, $1\le m\le 10$ or $m = 12$, and $({\mathbb Z}/2{\mathbb Z})\times ({\mathbb Z}/\nu{\mathbb Z})$, $1\le \nu\le 4$. The number $r$ is called the rank of the elliptic curve over ${\mathbb Q}$, or its ${\mathbb Q}$-rank. Examples are known of elliptic curves over ${\mathbb Q}$ of rank $\ge 12$. There is a conjecture (see [Ca], [Ta]) that over ${\mathbb Q}$ there exist elliptic curves of arbitrary large rank.

In the study of $X({\mathbb Q})$ one uses the Tate height ${\hat h}:X({\mathbb Q})\to {\mathbb R}^+$, which is a non-negative definite quadratic form on $X({\mathbb Q})$ (see [Ca], [Mu], [La], and also Height, in Diophantine geometry). For any $c\in {\mathbb R}^+$ the set $\{P\in X(\mathbb Q) | {\hat h}(P)\le c\}$ is finite. In particular, $\hat h$ vanishes precisely on the torsion subgroup of $X({\mathbb Q})_t$.

An important invariant of an elliptic curve is its Tate–Shafarevich group ${\rm Sha}\;(X)$ (see Weil–Châtelet group). The non-trivial elements of ${\rm Sha}\;(X)$, an elliptic curve without ${\mathbb Q}$-points, provide examples of elliptic curves for which the Hasse principle fails to hold. The group ${\rm Sha}\;(X)$ is periodic and for every $n$ the subgroup of its elements of order dividing $n$ is finite. For a large number of elliptic curves it has been verified that the 2- and $3$-components of ${\rm Sha}$ are finite (see [Ca], [CaFr], [Ma]). There is a conjecture that ${\rm Sha}$ is finite.

A conjecture of Birch and Swinnerton-Dyer asserts (see [Ma], [Ta]) that the order of the zero of the $L$-function $L(X,s)$ at $s=1$ is equal to the ${\mathbb Q}$-rank of $X$. In particular, $L(X,s)$ has a zero at $s=1$ if and only if $X({\mathbb Q})$ is infinite. So far (1984) the conjecture has not been proved for a single elliptic curve, but for elliptic curves with complex multiplication (and $j=1$) it has been established that when $X({\mathbb Q})$ is infinite, then the $L$-function has a zero at $s=1$ (see [CoWi]). The conjecture of Birch and Swinnerton-Dyer gives the principal term of the asymptotic expansion of the $L$-function as $s\to 1$; in it there occur the orders of the groups ${\rm Sha}\;(X)$ and $X({\mathbb Q})_t$ and the determinant of the Tate height [Ca]. It can be restated in terms of the Tamagawa numbers (cf. Tamagawa number, see [Bl]).

There is a conjecture of Weil that an elliptic curve $X$ has a uniformization by modular functions relative to the congruence subgroup $\Gamma_0(N)$ of the modular group $\Gamma$ (see [Ma] and also Zeta-function in algebraic geometry). This conjecture has been proved for elliptic functions with complex multiplication. It is known (see [Be]) that every algebraic curve over $\mathbb Q$ can be uniformized (cf. Uniformization) by modular functions relative to some subgroup of $\Gamma$ of finite index.

References

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How to Cite This Entry:
Elliptic curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_curve&oldid=19579
This article was adapted from an original article by Yu.G. ZarkhinVal.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article