Namespaces
Variants
Actions

Difference between revisions of "Elliptic curve"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (Reverted edits by Rehmann (talk) to last revision by 127.0.0.1)
 
(12 intermediate revisions by 3 users not shown)
Line 1: Line 1:
A non-singular complete [[Algebraic curve|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 integral]]; [[Elliptic function|Elliptic function]]).
+
{{MSC|14h57|11Gxx,14K15}}
 +
{{TEX|done}}
  
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.==
+
An ''elliptic curve'' is a non-singular complete
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354501.png" /> be an elliptic curve over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354502.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354503.png" /> is biregularly isomorphic to a plane cubic curve (see [[#References|[1]]], [[#References|[9]]], [[#References|[13]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354504.png" />, then in the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354505.png" /> there is an affine coordinate system in which the equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354506.png" /> is in normal Weierstrass form:
+
[[Algebraic curve|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 integral]];
 +
[[Elliptic function|Elliptic function]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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 curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354508.png" /> is non-singular if and only if the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354509.png" /> does not have multiple zeros, that is, if the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545010.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545011.png" /> the curve (1) has a unique point at infinity, which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545013.png" /> is a point of inflection of (1), and the tangent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545014.png" /> is the line at infinity. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545016.png" />-invariant of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545017.png" />,
+
==The geometry of an elliptic curve.==
 
+
Let $X$ be an elliptic curve
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545018.png" /></td> </tr></table>
+
over an algebraically closed field $k$. Then $X$ is biregularly
 
+
isomorphic to a plane cubic curve (see
does not depend on the choice of the coordinate system. Two elliptic curves have the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545019.png" />-invariant if and only if they are biregularly isomorphic. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545020.png" /> there is an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545021.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545022.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545023.png" />.
+
{{Cite|Ca}},
 +
{{Cite|La2}},
 +
{{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
 +
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.==
 
==The group structure on an elliptic curve.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545024.png" /> be a fixed point on an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545025.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545026.png" /> assigning to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545027.png" /> the [[Divisor|divisor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545029.png" /> establishes a one-to-one correspondence between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545030.png" /> and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545031.png" /> of divisor classes of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545033.png" />, that is, the [[Picard variety|Picard variety]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545034.png" />. This correspondence endows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545035.png" /> with the structure of an Abelian group that is compatible with the structure of an algebraic variety and that turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545036.png" /> into a one-dimensional Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545037.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545038.png" /> is the trivial element of the group. This group structure has the following geometric description. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545039.png" /> be a smooth plane cubic curve. Then the sum of two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545041.png" /> is defined by the rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545043.png" /> is the third point of intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545044.png" /> with the line passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545046.png" />. In other words, the sum of three points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545047.png" /> vanishes if and only if the points are collinear.
+
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|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|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.==
 
==An elliptic curve as a one-dimensional Abelian variety.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545048.png" /> denote the endomorphism of multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545050.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545051.png" /> is an elliptic curve with distinguished point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545052.png" />, then any rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545053.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545056.png" /> is a homomorphism of Abelian varieties. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545057.png" /> is either a constant mapping at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545058.png" /> or is an [[Isogeny|isogeny]], that is, there is a homomorphism of Abelian varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545061.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545062.png" /> (see [[#References|[1]]], [[#References|[6]]]).
+
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|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
 +
{{Cite|Ca}},
 +
{{Cite|Ha}}).
  
The automorphism group of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545063.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545064.png" />, and its subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545065.png" /> of automorphisms leaving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545066.png" /> fixed is non-trivial and finite. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545067.png" /> is not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545068.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545069.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545070.png" /> is neither 0 nor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545071.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545072.png" /> consists of the two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545074.png" />. The order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545075.png" /> is 4 when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545076.png" /> and 6 when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545077.png" /> (see [[#References|[1]]], [[#References|[6]]], [[#References|[13]]]).
+
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
 +
{{Cite|Ca}},
 +
{{Cite|Ha}},
 +
{{Cite|Ta}}).
  
An important invariant of an elliptic curve is the endomorphism ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545078.png" /> of the Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545079.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545080.png" /> defines an imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545081.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545083.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545084.png" /> is an elliptic curve with complex multiplication. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545085.png" /> can be of one of the following types (see [[#References|[1]]], [[#References|[9]]], [[#References|[13]]]): I) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545086.png" />; II) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545087.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545088.png" /> is the ring of algebraic integers of an imaginary quadratic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545090.png" />; or III) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545091.png" /> is a non-commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545092.png" />-algebra of rank 4 without divisors of zero. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545094.png" /> is a maximal order in the quaternion algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545095.png" /> ramified only at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545097.png" />. Such elliptic curves exist for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545098.png" /> and are called supersingular; elliptic curves in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545099.png" /> that are not supersingular are said to be ordinary.
+
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
 +
{{Cite|Ca}},
 +
{{Cite|La2}},
 +
{{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 $k$ 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450100.png" /> of points of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450101.png" /> with orders that divide <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450102.png" /> has the following structure: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450103.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450104.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450105.png" /> and ordinary elliptic curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450106.png" />, while for supersingular elliptic curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450107.png" />. For a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450108.png" /> the [[Tate module|Tate module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450109.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450110.png" />.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450111.png" /> be an elliptic curve over an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450112.png" />. If the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450113.png" />-rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450114.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450115.png" /> is not empty, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450116.png" /> is biregularly isomorphic to a plane cubic curve (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450117.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450118.png" />). The point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450119.png" /> of (1) is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450120.png" />. As above, one can introduce a group structure on (1), turning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450121.png" /> into a one-dimensional Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450122.png" /> and turning the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450123.png" /> into an Abelian group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450124.png" /> as trivial element. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450125.png" /> is finitely generated over its prime subfield, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450126.png" /> is a finitely-generated group (the Mordell–Weil theorem).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450127.png" /> there is defined the [[Jacobi variety|Jacobi variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450128.png" />, which is a one-dimensional Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450129.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450130.png" /> is a [[Principal homogeneous space|principal homogeneous space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450131.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450132.png" /> is not empty, then the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450133.png" /> specifies an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450134.png" /> under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450135.png" /> becomes the trivial element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450136.png" />. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450137.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450138.png" /> are isomorphic over a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450139.png" /> (see [[#References|[1]]], [[#References|[4]]], [[#References|[13]]]).
+
For any elliptic curve $X$ there is defined the
 +
[[Jacobi variety|Jacobi variety]] $J(X)$, which is a one-dimensional
 +
Abelian variety over $k$, and $X$ is a
 +
[[Principal homogeneous space|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
 +
{{Cite|Ca}},
 +
{{Cite|CaFr}},
 +
{{Cite|Ta}}).
  
 
==Elliptic curves over the field of complex numbers.==
 
==Elliptic curves over the field of complex numbers.==
An elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450140.png" /> is a compact [[Riemann surface|Riemann surface]] of genus 1, and vice versa. The group structure turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450141.png" /> into a complex Lie group, which is a one-dimensional complex torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450142.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450143.png" /> is a lattice in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450144.png" />. Conversely, any one-dimensional complex torus is an elliptic curve (see [[#References|[3]]]). From the topological point of view, an elliptic curve is a two-dimensional torus.
+
An elliptic
 +
curve over ${\mathbb C}$ is a compact
 +
[[Riemann surface|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
 +
{{Cite|Mu}}). From the topological point of view, an elliptic
 +
curve is a two-dimensional torus.
  
The theory of elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450145.png" /> is in essence equivalent to the theory of elliptic functions. An identification of a torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450146.png" /> with an elliptic curve can be effected as follows. The elliptic functions with a given period lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450147.png" /> form a field generated by the Weierstrass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450148.png" />-function (see [[Weierstrass elliptic functions|Weierstrass elliptic functions]]) and its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450149.png" />, which are connected by the relation
+
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|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 $X\simeq J(X)$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450150.png" /></td> </tr></table>
+
The description of the set of all elliptic curves as tori ${\mathbb C}/\Lambda$ leads to
 +
the
 +
[[Modular function|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|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 $j(X)=1728J(\tau)$.
  
The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450151.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450152.png" />) induces an isomorphism between the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450153.png" /> and the elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450154.png" /> with equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450155.png" />. The identification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450156.png" /> given by (1) with the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450157.png" /> is effected by curvilinear integrals of the holomorphic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450158.png" /> and gives an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450159.png" />.
+
An elliptic curve $X$ has complex multiplication if and only if $\tau$ is
 
+
an imaginary
The description of the set of all elliptic curves as tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450160.png" /> leads to the [[Modular function|modular function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450161.png" />. Two lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450162.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450163.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450164.png" /> is generated by the numbers 1 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450165.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450166.png" />. Two lattices with bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450167.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450168.png" /> are similar if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450169.png" /> for an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450170.png" /> of the [[Modular group|modular group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450171.png" />. The modular function
+
[[Quadratic irrationality|quadratic irrationality]]. In this case ${\mathbb R}$
 
+
is a subring of finite index in the ring of algebraic integers of the
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450172.png" /></td> </tr></table>
+
imaginary quadratic field ${\mathbb Q}(\tau)$. Elliptic curves with complex
 
+
multiplication are closely connected with the
is also called the absolute invariant; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450173.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450174.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450175.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450176.png" /> produces a one-to-one correspondence between the classes of isomorphic elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450177.png" /> and the complex numbers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450178.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450179.png" />.
+
[[Class field theory|class field theory]] for imaginary quadratic
 
+
fields (see
An elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450180.png" /> has complex multiplication if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450181.png" /> is an imaginary [[Quadratic irrationality|quadratic irrationality]]. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450182.png" /> is a subring of finite index in the ring of algebraic integers of the imaginary quadratic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450183.png" />. Elliptic curves with complex multiplication are closely connected with the [[Class field theory|class field theory]] for imaginary quadratic fields (see [[#References|[4]]], [[#References|[8]]]).
+
{{Cite|CaFr}},
 +
{{Cite|La}}).
  
 
==Arithmetic of elliptic curves.==
 
==Arithmetic of elliptic curves.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450184.png" /> be an elliptic curve over the finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450185.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450186.png" /> elements. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450187.png" /> is always non-empty and finite. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450188.png" /> is endowed with the structure of a one-dimensional Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450189.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450190.png" /> with that of a finite Abelian group. The order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450191.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450192.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450193.png" />. The characteristic polynomial of the [[Frobenius endomorphism|Frobenius endomorphism]] acting on the Tate module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450194.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450195.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450196.png" />. Its roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450197.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450198.png" /> are complex-conjugate algebraic integers of modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450199.png" />. For any finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450200.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450201.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450202.png" />, the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450203.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450204.png" />. The [[Zeta-function|zeta-function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450205.png" /> is
+
Let $X$ be an elliptic curve over
 
+
the finite field $k$ with $q$ elements. The set $X(k)$ is always
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450206.png" /></td> </tr></table>
+
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
For any algebraic integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450207.png" /> of modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450208.png" /> in some imaginary quadratic field (or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450209.png" />) one can find an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450210.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450211.png" /> such that the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450212.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450213.png" />.
+
finite Abelian group. The order $A$ of $X(k)$ satisfies $|q+1-A|\le 2 \sqrt{q}$. The
 
+
characteristic polynomial of the
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450214.png" /> be the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450215.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450216.png" />-adic numbers or a finite algebraic extension of it, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450217.png" /> be the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450218.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450219.png" /> be an elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450220.png" />, and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450221.png" /> is non-empty. The group structure turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450222.png" /> into a commutative compact one-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450223.png" />-adic Lie group (cf. [[Lie-group-adic|Lie group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450224.png" />-adic]]). The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450225.png" /> is Pontryagin-dual to the [[Weil–Châtelet group|Weil–Châtelet group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450226.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450227.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450228.png" /> is a Tate curve (see [[#References|[1]]], [[#References|[5]]]) and there exists a canonical uniformization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450229.png" /> analogous to the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450230.png" />.
+
[[Frobenius endomorphism|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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450231.png" /> be an elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450232.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450233.png" /> is not empty. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450234.png" /> is biregularly isomorphic to the curve (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450235.png" />. Of all curves of the form (1) that are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450236.png" /> with integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450237.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450238.png" />, one chooses the one for which the absolute value of the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450239.png" /> is minimal. The conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450240.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450242.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450243.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450244.png" /> are defined as formal products of local factors:
+
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|Lie group, $p$-adic]]). The group $X(k)$ is
 +
Pontryagin-dual to the
 +
[[Weil–Châtelet group|Weil–Châtelet group]] ${\rm WC}(k,X)$. If $j(X)\notin B$, then $X$ is a
 +
Tate curve (see
 +
{{Cite|Ca}},
 +
{{Cite|Ma}}) and there exists a canonical uniformization of
 +
$X(k)$ analogous to the case of ${\mathbb C}$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450245.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
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
 +
{{Cite|Ca}},
 +
{{Cite|Ma}},
 +
{{Cite|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
  
over all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450246.png" /> (see [[#References|[1]]], [[#References|[5]]], [[#References|[13]]]). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450247.png" /> is some power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450248.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450249.png" /> is a meromorphic function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450250.png" /> that has neither a zero nor a pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450251.png" />. To determine the local factors one considers the reduction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450252.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450253.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450254.png" />), which is a plane projective curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450255.png" /> over the residue class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450256.png" /> and is given in an affine coordinate system by the equation
+
$$\xi_X(s) = N^{s/2}(2\pi)^{-s}\;\Gamma(s)L(X,s)$$
 +
(where $\Gamma(s)$ is the
 +
[[Gamma-function|gamma-function]]) satisfies the functional equation
 +
$\xi_X(s) = W\xi_X(2-s)$ with $W=\pm1$ (see
 +
{{Cite|Ma}},
 +
{{Cite|Mu}}). This conjecture has been proved for elliptic
 +
curves with complex multiplication.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450257.png" /></td> </tr></table>
+
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
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450258.png" /> be the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450259.png" />-points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450260.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450261.png" /> does not divide <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450262.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450263.png" /> is an elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450264.png" />, and one puts
+
$r$. $X({\mathbb Q})_t$ is isomorphic to one of the following 15 groups (see
 
+
{{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 $r$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450265.png" /></td> </tr></table>
+
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
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450266.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450267.png" />, then the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450268.png" /> has a multiple root, and one puts
+
$\ge 12$. There is a conjecture (see
 
+
{{Cite|Ca}},
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450269.png" /></td> </tr></table>
+
{{Cite|Ta}}) that over ${\mathbb Q}$ there exist elliptic curves of
 
+
arbitrary large rank.
(depending on whether it is a triple or a double root). The product (2) converges in the right half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450270.png" />. It has been conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450271.png" /> has a meromorphic extension to the whole complex plane and that the function
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450272.png" /></td> </tr></table>
 
 
 
(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450273.png" /> is the [[Gamma-function|gamma-function]]) satisfies the functional equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450275.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450276.png" /> (see [[#References|[5]]], [[#References|[3]]]). This conjecture has been proved for elliptic curves with complex multiplication.
 
 
 
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450277.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450278.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450279.png" /> is a finite Abelian group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450280.png" /> is a free Abelian group of a certain finite rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450281.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450282.png" /> is isomorphic to one of the following 15 groups (see [[#References|[11]]]): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450283.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450284.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450285.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450286.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450287.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450288.png" /> is called the rank of the elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450289.png" />, or its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450291.png" />-rank. Examples are known of elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450292.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450293.png" />. There is a conjecture (see [[#References|[1]]], [[#References|[13]]]) that over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450294.png" /> there exist elliptic curves of arbitrary large rank.
 
 
 
In the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450295.png" /> one uses the Tate height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450296.png" />, which is a non-negative definite quadratic form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450297.png" /> (see [[#References|[1]]], [[#References|[3]]], [[#References|[8]]], and also [[Height, in Diophantine geometry|Height, in Diophantine geometry]]). For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450298.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450299.png" /> is finite. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450300.png" /> vanishes precisely on the torsion subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450301.png" />.
 
 
 
An important invariant of an elliptic curve is its Tate–Shafarevich group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450302.png" /> (see [[Weil–Châtelet group|Weil–Châtelet group]]). The non-trivial elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450303.png" />, an elliptic curve without <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450304.png" />-points, provide examples of elliptic curves for which the [[Hasse principle|Hasse principle]] fails to hold. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450305.png" /> is periodic and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450306.png" /> the subgroup of its elements of order dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450307.png" /> is finite. For a large number of elliptic curves it has been verified that the 2- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450308.png" />-components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450309.png" /> are finite (see [[#References|[1]]], [[#References|[4]]], [[#References|[5]]]). There is a conjecture that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450310.png" /> is finite.
 
 
 
A conjecture of Birch and Swinnerton-Dyer asserts (see [[#References|[5]]], [[#References|[13]]]) that the order of the zero of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450311.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450312.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450313.png" /> is equal to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450314.png" />-rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450315.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450316.png" /> has a zero at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450317.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450318.png" /> is infinite. So far (1984) the conjecture has not been proved for a single elliptic curve, but for elliptic curves with complex multiplication (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450319.png" />) it has been established that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450320.png" /> is infinite, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450321.png" />-function has a zero at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450322.png" /> (see [[#References|[14]]]). The conjecture of Birch and Swinnerton-Dyer gives the principal term of the asymptotic expansion of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450323.png" />-function as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450324.png" />; in it there occur the orders of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450325.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450326.png" /> and the determinant of the Tate height [[#References|[1]]]. It can be restated in terms of the Tamagawa numbers (cf. [[Tamagawa number|Tamagawa number]], see [[#References|[7]]]).
 
 
 
There is a conjecture of Weil that an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450327.png" /> has a uniformization by modular functions relative to the congruence subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450328.png" /> of the modular group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450329.png" /> (see [[#References|[5]]] and also [[Zeta-function|Zeta-function]] in algebraic geometry). This conjecture has been proved for elliptic functions with complex multiplication. It is known (see [[#References|[15]]]) that every algebraic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450330.png" /> can be uniformized (cf. [[Uniformization|Uniformization]]) by modular functions relative to some subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450331.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450332.png" />"  ''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>
 
  
 +
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
 +
{{Cite|Ca}},
 +
{{Cite|Mu}},
 +
{{Cite|La}}, and also
 +
[[Height, in Diophantine geometry|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|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|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
 +
{{Cite|Ca}},
 +
{{Cite|CaFr}},
 +
{{Cite|Ma}}). There is a conjecture that ${\rm Sha}$ is finite.
  
====Comments====
+
A conjecture of Birch and Swinnerton-Dyer asserts (see
 +
{{Cite|Ma}},
 +
{{Cite|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
 +
{{Cite|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
 +
{{Cite|Ca}}. It can be restated in terms of the Tamagawa
 +
numbers (cf.
 +
[[Tamagawa number|Tamagawa number]], see
 +
{{Cite|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
 +
{{Cite|Ma}} and also
 +
[[Zeta-function|Zeta-function]] in algebraic geometry). This
 +
conjecture has been proved for elliptic functions with complex
 +
multiplication. It is known (see
 +
{{Cite|Be}}) that every algebraic curve over $\mathbb Q$ can be
 +
uniformized (cf.
 +
[[Uniformization|Uniformization]]) by modular functions relative to
 +
some subgroup of $\Gamma$ of finite index.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Mazur,   "Modular curves and the Eisenstein ideal" ''Publ. Math. IHES'' , '''47'''  (1978pp. 33–186</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.H. Silverman,   "The arithmetic of elliptic curves" , Springer  (1986)</TD></TR></table>
+
{|
 +
|-
 +
|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"|{{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

[Be] 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
[Bl] S. Bloch, "A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture" Invent. Math., 58 (1980) pp. 65–76 MR0570874 Zbl 0444.14015
[Ca] J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc., 41 (1966) pp. 193–291 MR0199150
[CaFr] J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967) MR0215665 Zbl 0153.07403
[CoWi] J. Coates, A. Wiles, "On the conjecture of Birch and Swinnerton-Dyer" Invent. Math., 39 (1977) pp. 223–251 MR0463176 Zbl 0359.14009
[Ha] R. Hartshorne, "Algebraic geometry", Springer (1977) pp. 91 MR0463157 Zbl 0367.14001
[HuCo] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen", Springer (1964) MR0173749 Zbl 0135.12101
[La] S. Lang, "Elliptic curves: Diophantine analysis", Springer (1978) MR0518817 Zbl 0388.10001
[La2] S. Lang, "Elliptic functions", Addison-Wesley (1973) MR0409362 Zbl 0316.14001
[Ma] 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 MR0401653
[Ma2] B. Mazur, "Rational isogenies of prime degree" Invent. Math., 44 (1978) pp. 129–162 MR0482230 Zbl 0386.14009
[Ma3] B. Mazur, "Modular curves and the Eisenstein ideal" Publ. Math. IHES, 47 (1977) pp. 33–186 MR0488287 Zbl 0394.14008
[Me] J.F. Mestre, "Construction d'une courbe elliptique de rang $\ge 12$" C.R. Acad. Sci. Paris Sér. 1, 295 (1982) pp. 643–644 MR0688896 Zbl 0541.14027
[Mu] D. Mumford, "Abelian varieties", Oxford Univ. Press (1974) Zbl 0326.14012
[SeDeKu] J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.), Modular functions of one variable. 4, Lect. notes in math., 476, Springer (1975) MR0404145 MR0404146
[Si] J.H. Silverman, "The arithmetic of elliptic curves", Springer (1986) MR0817210 Zbl 0585.14026
[Ta] J. Tate, "The arithmetic of elliptic curves" Invent. Math., 23 (1974) pp. 197–206 MR0419359 Zbl 0296.14018
How to Cite This Entry:
Elliptic curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_curve&oldid=19580
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