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A non-singular complete
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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]]).
[[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]]).
 
  
Examples. A non-singular plane projective cubic curve; the
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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.
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.==
 
==The geometry of an elliptic curve.==
Let $X$ be an elliptic curve
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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:
over an algebraically closed field $k$. Then $X$ is biregularly
+
 
isomorphic to a plane cubic curve (see
+
<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>
[[#References|[1]]],
+
 
[[#References|[9]]],
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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" />,
[[#References|[13]]]). If ${\rm char k} \ne 2,3$, then in the projective plane ${\mathbb P}^2$ there
+
 
is an affine coordinate system in which the equation of $X$ is in
+
<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>
normal Weierstrass form:  
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$$y^2=x^3+ax+b$$
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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" />.
The curve $ $ is non-singular if and
 
only if the polynomial $_$ does not have multiple zeros, that is, if
 
the discriminant $_$. In $_$ the curve (1) has a unique point at
 
infinity, which is denoted by $_$; $_$ is a point of inflection of
 
(1), and the tangent at $_$ is the line at infinity. The $_$-invariant
 
of an elliptic curve $_$,  
 
$$_$$
 
does not depend on the choice of the
 
coordinate system. Two elliptic curves have the same $_$-invariant if
 
and only if they are biregularly isomorphic. For any $_$ there is an
 
elliptic curve $_$ over $_$ with $_$.
 
  
 
==The group structure on an elliptic curve.==
 
==The group structure on an elliptic curve.==
Let $_$ be a fixed point
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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.
on an elliptic curve $_$. The mapping $_$ assigning to a point $_$ the
 
[[Divisor|divisor]] $_$ on $_$ establishes a one-to-one correspondence
 
between $_$ and the group $_$ of divisor classes of degree $_$ on $_$,
 
that is, the
 
[[Picard variety|Picard variety]] of $_$. This correspondence endows
 
$_$ with the structure of an Abelian group that is compatible with the
 
structure of an algebraic variety and that turns $_$ into a
 
one-dimensional Abelian variety $_$; here $_$ is the trivial element
 
of the group. This group structure has the following geometric
 
description. Let $_$ be a smooth plane cubic curve. Then the sum of
 
two points $_$ and $_$ is defined by the rule $_$, where $_$ is the
 
third point of intersection of $_$ with the line passing through $_$
 
and $_$. In other words, the sum of three points on $_$ 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 $_$
+
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]]]).
denote the endomorphism of multiplication by $_$ in $_$. If $_$ is an
 
elliptic curve with distinguished point $_$, then any rational mapping
 
$_$ has the form $_$, where $_$ and $_$ is a homomorphism of Abelian
 
varieties. Here $_$ is either a constant mapping at $_$ or is an
 
[[Isogeny|isogeny]], that is, there is a homomorphism of Abelian
 
varieties $_$ such that $_$ and $_$ for some $_$ (see
 
[[#References|[1]]],
 
[[#References|[6]]]).
 
  
The automorphism group of an elliptic curve $_$ acts transitively on
+
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]]]).
$_$, and its subgroup $_$ of automorphisms leaving $_$ fixed is
 
non-trivial and finite. Suppose that $_$ is not $_$ or $_$. When $_$
 
is neither 0 nor $_$, then $_$ consists of the two elements $_$ and
 
$_$. The order of $_$ is 4 when $_$ and 6 when $_$ (see
 
[[#References|[1]]],
 
[[#References|[6]]],
 
[[#References|[13]]]).
 
  
An important invariant of an elliptic curve is the endomorphism ring
+
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.
$_$ of the Abelian variety $_$. The mapping $_$ defines an imbedding
 
of $_$ in $_$. If $_$, one says that $_$ is an elliptic curve with
 
complex multiplication. The ring $_$ can be of one of the following
 
types (see
 
[[#References|[1]]],
 
[[#References|[9]]],
 
[[#References|[13]]]): I) $_$; II) $_$, where $_$ is the ring of
 
algebraic integers of an imaginary quadratic field $_$ and $_$; or
 
III) $_$ is a non-commutative $_$-algebra of rank 4 without divisors
 
of zero. In this case $_$ and $_$ is a maximal order in the quaternion
 
algebra over $_$ ramified only at $_$ and $_$. Such elliptic curves
 
exist for all $_$ and are called supersingular; elliptic curves in
 
characteristic $_$ that are not supersingular are said to be ordinary.
 
  
The group $_$ of points of an elliptic curve $_$ with orders that
+
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" />.
divide $_$ has the following structure: $_$ when $_$. For $_$ and
 
ordinary elliptic curves $_$, while for supersingular elliptic curves
 
$_$. For a prime number $_$ the
 
[[Tate module|Tate module]] $_$ is isomorphic to $_$.
 
  
 
==Elliptic curves over non-closed fields.==
 
==Elliptic curves over non-closed fields.==
Let $_$ be an elliptic
+
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).
curve over an arbitrary field $_$. If the set of $_$-rational points
 
$_$ of $_$ is not empty, then $_$ is biregularly isomorphic to a plane
 
cubic curve (1) with $_$ ($_$). The point at infinity $_$ of (1) is
 
defined over $_$. As above, one can introduce a group structure on
 
(1), turning $_$ into a one-dimensional Abelian variety over $_$ and
 
turning the set $_$ into an Abelian group with $_$ as trivial
 
element. If $_$ is finitely generated over its prime subfield, then
 
$_$ is a finitely-generated group (the Mordell–Weil theorem).
 
  
For any elliptic curve $_$ there is defined the
+
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]]]).
[[Jacobi variety|Jacobi variety]] $_$, which is a one-dimensional
 
Abelian variety over $_$, and $_$ is a
 
[[Principal homogeneous space|principal homogeneous space]] over
 
$_$. If $_$ is not empty, then the choice of $_$ specifies an
 
isomorphism $_$ under which $_$ becomes the trivial element of $_$. In
 
general, $_$ and $_$ are isomorphic over a finite extension of $_$
 
(see
 
[[#References|[1]]],
 
[[#References|[4]]],
 
[[#References|[13]]]).
 
  
 
==Elliptic curves over the field of complex numbers.==
 
==Elliptic curves over the field of complex numbers.==
An elliptic
+
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.
curve over $_$ is a compact
+
 
[[Riemann surface|Riemann surface]] of genus 1, and vice versa. The
+
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
group structure turns $_$ into a complex Lie group, which is a
+
 
one-dimensional complex torus $_$, where $_$ is a lattice in the
+
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complex plane $_$. 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.
 
  
The theory of elliptic curves over $_$ is in essence equivalent to the
+
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" />.
theory of elliptic functions. An identification of a torus $_$ with an
 
elliptic curve can be effected as follows. The elliptic functions with
 
a given period lattice $_$ form a field generated by the Weierstrass
 
$_$-function (see
 
[[Weierstrass elliptic functions|Weierstrass elliptic functions]]) and
 
its derivative $_$, which are connected by the relation
 
$$_$$
 
The
 
mapping $_$ ($_$) induces an isomorphism between the torus $_$ and the
 
elliptic curve $_$ with equation $_$. The identification of $_$ given
 
by (1) with the torus $_$ is effected by curvilinear integrals of the
 
holomorphic form $_$ and gives an isomorphism $_$.
 
  
The description of the set of all elliptic curves as tori $_$ leads to
+
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
the
 
[[Modular function|modular function]] $_$. Two lattices $_$ and $_$
 
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 $_$ is generated by the
 
numbers 1 and $_$ in $_$. Two lattices with bases $_$ and $_$ are
 
similar if and only if $_$ for an element $_$ of the
 
[[Modular group|modular group]] $_$. The modular function  
 
$$_$$
 
is
 
also called the absolute invariant; $_$ if and only if $_$ for some
 
$_$, and the function $_$ produces a one-to-one correspondence between
 
the classes of isomorphic elliptic curves over $_$ and the complex
 
numbers. If $_$, then $_$.
 
  
An elliptic curve $_$ has complex multiplication if and only if $_$ is
+
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an imaginary
+
 
[[Quadratic irrationality|quadratic irrationality]]. In this case $_$
+
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" />.
is a subring of finite index in the ring of algebraic integers of the
+
 
imaginary quadratic field $_$. Elliptic curves with complex
+
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]]]).
multiplication are closely connected with the
 
[[Class field theory|class field theory]] for imaginary quadratic
 
fields (see
 
[[#References|[4]]],
 
[[#References|[8]]]).
 
  
 
==Arithmetic of elliptic curves.==
 
==Arithmetic of elliptic curves.==
Let $_$ be an elliptic curve over
+
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
the finite field $_$ with $_$ elements. The set $_$ is always
+
 
non-empty and finite. Hence $_$ is endowed with the structure of a
+
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one-dimensional Abelian variety over $_$, and $_$ with that of a
+
 
finite Abelian group. The order $_$ of $_$ satisfies $_$. The
+
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" />.
characteristic polynomial of the
+
 
[[Frobenius endomorphism|Frobenius endomorphism]] acting on the Tate
+
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" />.
module $_$, $_$, is $_$. Its roots $_$ and $_$ are complex-conjugate
+
 
algebraic integers of modulus $_$. For any finite extension $_$ of $_$
+
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:
of degree $_$, the order of $_$ is $_$. The
+
 
[[Zeta-function|zeta-function]] of $_$ is  
+
<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>
$$_$$
+
 
For any algebraic
+
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
integer $_$ of modulus $_$ in some imaginary quadratic field (or in
+
 
$_$) one can find an elliptic curve $_$ over $_$ such that the order
+
<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>
of $_$ is $_$.
+
 
 +
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
 +
 
 +
<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>
 +
 
 +
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
 +
 
 +
<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>
  
Let $_$ be the field $_$ of $_$-adic numbers or a finite algebraic
+
(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
extension of it, let $_$ be the ring of integers of $_$, let $_$ be an
 
elliptic curve over $_$, and suppose that $_$ is non-empty. The group
 
structure turns $_$ into a commutative compact one-dimensional
 
$_$-adic Lie group (cf.
 
[[Lie-group-adic|Lie group, $_$-adic]]). The group $_$ is
 
Pontryagin-dual to the
 
[[Weil–Châtelet group|Weil–Châtelet group]] $_$. If $_$, then $_$ is a
 
Tate curve (see
 
[[#References|[1]]],
 
[[#References|[5]]]) and there exists a canonical uniformization of
 
$_$ analogous to the case of $_$.
 
  
Let $_$ be an elliptic curve over $_$ for which $_$ is not empty. Then
+
<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>
$_$ is biregularly isomorphic to the curve (1) with $_$. Of all curves
 
of the form (1) that are isomorphic to $_$ with integers $_$ and $_$,
 
one chooses the one for which the absolute value of the discriminant
 
$_$ is minimal. The conductor $_$ and the $_$-function $_$ of $_$ are
 
defined as formal products of local factors:  
 
$$_$$
 
over all prime
 
numbers $_$ (see
 
[[#References|[1]]],
 
[[#References|[5]]],
 
[[#References|[13]]]). Here $_$ is some power of $_$, and $_$ is a
 
meromorphic function of the complex variable $_$ that has neither a
 
zero nor a pole at $_$. To determine the local factors one considers
 
the reduction of $_$ modulo $_$ ($_$), which is a plane projective
 
curve $_$ over the residue class field $_$ and is given in an affine
 
coordinate system by the equation
 
$$_$$
 
Let $_$ be the number of
 
$_$-points on $_$. If $_$ does not divide $_$, then $_$ is an elliptic
 
curve over $_$, and one puts
 
$$_$$
 
If $_$ divides $_$, then the
 
polynomial $_$ has a multiple root, and one puts
 
$$_$$
 
(depending on
 
whether it is a triple or a double root). The product (2) converges in
 
the right half-plane $_$. It has been conjectured that $_$ has a
 
meromorphic extension to the whole complex plane and that the function
 
  
$$_$$
+
(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.
(where $_$ is the
 
[[Gamma-function|gamma-function]]) satisfies the functional equation
 
$_$ with $_$ (see
 
[[#References|[5]]],
 
[[#References|[3]]]). This conjecture has been proved for elliptic
 
curves with complex multiplication.
 
  
The group $_$ is isomorphic to $_$, where $_$ is a finite Abelian
+
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.
group and $_$ is a free Abelian group of a certain finite rank
 
$_$. $_$ is isomorphic to one of the following 15 groups (see
 
[[#References|[11]]]): $_$, $_$ or $_$, and $_$, $_$. The number $_$
 
is called the rank of the elliptic curve over $_$, or its
 
$_$-rank. Examples are known of elliptic curves over $_$ of rank
 
$_$. There is a conjecture (see
 
[[#References|[1]]],
 
[[#References|[13]]]) that over $_$ there exist elliptic curves of
 
arbitrary large rank.
 
  
In the study of $_$ one uses the Tate height $_$, which is a
+
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" />.
non-negative definite quadratic form on $_$ (see
 
[[#References|[1]]],
 
[[#References|[3]]],
 
[[#References|[8]]], and also
 
[[Height, in Diophantine geometry|Height, in Diophantine
 
geometry]]). For any $_$ the set $_$ is finite. In particular, $_$
 
vanishes precisely on the torsion subgroup of $_$.
 
  
An important invariant of an elliptic curve is its Tate–Shafarevich
+
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.
group $_$ (see
 
[[Weil–Châtelet group|Weil–Châtelet group]]). The non-trivial elements
 
of $_$, an elliptic curve without $_$-points, provide examples of
 
elliptic curves for which the
 
[[Hasse principle|Hasse principle]] fails to hold. The group $_$ is
 
periodic and for every $_$ the subgroup of its elements of order
 
dividing $_$ is finite. For a large number of elliptic curves it has
 
been verified that the 2- and $_$-components of $_$ are finite (see
 
[[#References|[1]]],
 
[[#References|[4]]],
 
[[#References|[5]]]). There is a conjecture that $_$ is finite.
 
  
A conjecture of Birch and Swinnerton-Dyer asserts (see
+
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]]]).
[[#References|[5]]],
 
[[#References|[13]]]) that the order of the zero of the $_$-function
 
$_$ at $_$ is equal to the $_$-rank of $_$. In particular, $_$ has a
 
zero at $_$ if and only if $_$ is infinite. So far (1984) the
 
conjecture has not been proved for a single elliptic curve, but for
 
elliptic curves with complex multiplication (and $_$) it has been
 
established that when $_$ is infinite, then the $_$-function has a
 
zero at $_$ (see
 
[[#References|[14]]]). The conjecture of Birch and Swinnerton-Dyer
 
gives the principal term of the asymptotic expansion of the
 
$_$-function as $_$; in it there occur the orders of the groups $_$
 
and $_$ 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 $_$ has a
+
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.
uniformization by modular functions relative to the congruence
 
subgroup $_$ of the modular group $_$ (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 $_$ can be
 
uniformized (cf.
 
[[Uniformization|Uniformization]]) by modular functions relative to
 
some subgroup of $_$ of finite index.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD
+
<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>
valign="top"> J.W.S. Cassels, "Diophantine equations with special
 
reference to elliptic curves" ''J. London Math. Soc.'' , '''41'''
 
(1966) pp. 193–291</TD></TR><TR><TD valign="top">[2]</TD> <TD
 
valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine
 
Funktionentheorie und elliptische Funktionen" , Springer
 
(1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
 
D. Mumford, "Abelian varieties" , Oxford Univ. Press
 
(1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
 
J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory''
 
, Acad. Press (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD
 
valign="top"> Yu.I. Manin, "Cyclotomic fields and modular curves"
 
''Russian Math. Surveys'' , '''26''' : 6 (1971) pp. 6–78 ''Uspekhi
 
Mat. Nauk'' , '''26''' : 6 (1971) pp. 7–71</TD></TR><TR><TD
 
valign="top">[6]</TD> <TD valign="top"> R. Hartshorne, "Algebraic
 
geometry" , Springer (1977) pp. 91</TD></TR><TR><TD
 
valign="top">[7]</TD> <TD valign="top"> S. Bloch, "A note on height
 
pairings, Tamagawa numbers, and the Birch and Swinnnerton-Dyer
 
conjecture" ''Invent. Math.'' , '''58''' (1980)
 
pp. 65–76</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">
 
S. Lang, "Elliptic curves; Diophantine analysis" , Springer
 
(1978)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">
 
S. Lang, "Elliptic functions" , Addison-Wesley (1973)</TD></TR><TR><TD
 
valign="top">[10]</TD> <TD valign="top"> B. Mazur, "Rational isogenies
 
of prime degree" ''Invent. Math.'' , '''44''' (1978)
 
pp. 129–162</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">
 
J.-P. Serre (ed.)  P. Deligne (ed.)  W. Kuyk (ed.) , ''Modular
 
functions of one variable. 4'' , ''Lect. notes in math.'' , '''476'''
 
, Springer (1975)</TD></TR><TR><TD valign="top">[12]</TD> <TD
 
valign="top"> J.F. Mestre, "Construction d'une courbe elliptique de
 
rang $_$" ''C.R. Acad. Sci. Paris Sér. 1'' , '''295''' (1982)
 
pp. 643–644</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">
 
J. Tate, "The arithmetic of elliptic curves" ''Invent. Math.'' ,
 
'''23''' (1974) pp. 197–206</TD></TR><TR><TD valign="top">[14]</TD>
 
<TD valign="top"> J. Coates, A. Wiles, "On the conjecture of Birch and
 
Swinnerton-Dyer" ''Invent. Math.'' , '''39''' (1977)
 
pp. 223–251</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">
 
G.V. Belyi, "On Galois extensions of a maximal cyclotomic field"
 
''Math. USSR Izv.'' , '''14''' : 2 (1980) pp. 247–256
 
''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''43''' (1979)
 
pp. 267–276</TD></TR></table>
 
  
  
Line 350: Line 98:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Mazur,   "Modular curves and the Eisenstein ideal" ''Publ. Math. IHES'' , '''47''' (1978) pp. 33–186</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.H. Silverman,   "The arithmetic of elliptic curves" , Springer (1986)</TD></TR></table>
valign="top"> B. Mazur, "Modular curves and the Eisenstein ideal"
 
''Publ. Math. IHES'' , '''47''' (1978) pp. 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>
 

Revision as of 14:33, 12 September 2011

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 be an elliptic curve over an algebraically closed field . Then is biregularly isomorphic to a plane cubic curve (see [1], [9], [13]). If , then in the projective plane there is an affine coordinate system in which the equation of is in normal Weierstrass form:

(1)

The curve is non-singular if and only if the polynomial does not have multiple zeros, that is, if the discriminant . In the curve (1) has a unique point at infinity, which is denoted by ; is a point of inflection of (1), and the tangent at is the line at infinity. The -invariant of an elliptic curve ,

does not depend on the choice of the coordinate system. Two elliptic curves have the same -invariant if and only if they are biregularly isomorphic. For any there is an elliptic curve over with .

The group structure on an elliptic curve.

Let be a fixed point on an elliptic curve . The mapping assigning to a point the divisor on establishes a one-to-one correspondence between and the group of divisor classes of degree on , that is, the Picard variety of . This correspondence endows with the structure of an Abelian group that is compatible with the structure of an algebraic variety and that turns into a one-dimensional Abelian variety ; here is the trivial element of the group. This group structure has the following geometric description. Let be a smooth plane cubic curve. Then the sum of two points and is defined by the rule , where is the third point of intersection of with the line passing through and . In other words, the sum of three points on vanishes if and only if the points are collinear.

An elliptic curve as a one-dimensional Abelian variety.

Let denote the endomorphism of multiplication by in . If is an elliptic curve with distinguished point , then any rational mapping has the form , where and is a homomorphism of Abelian varieties. Here is either a constant mapping at or is an isogeny, that is, there is a homomorphism of Abelian varieties such that and for some (see [1], [6]).

The automorphism group of an elliptic curve acts transitively on , and its subgroup of automorphisms leaving fixed is non-trivial and finite. Suppose that is not or . When is neither 0 nor , then consists of the two elements and . The order of is 4 when and 6 when (see [1], [6], [13]).

An important invariant of an elliptic curve is the endomorphism ring of the Abelian variety . The mapping defines an imbedding of in . If , one says that is an elliptic curve with complex multiplication. The ring can be of one of the following types (see [1], [9], [13]): I) ; II) , where is the ring of algebraic integers of an imaginary quadratic field and ; or III) is a non-commutative -algebra of rank 4 without divisors of zero. In this case and is a maximal order in the quaternion algebra over ramified only at and . Such elliptic curves exist for all and are called supersingular; elliptic curves in characteristic that are not supersingular are said to be ordinary.

The group of points of an elliptic curve with orders that divide has the following structure: when . For and ordinary elliptic curves , while for supersingular elliptic curves . For a prime number the Tate module is isomorphic to .

Elliptic curves over non-closed fields.

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

For any elliptic curve there is defined the Jacobi variety , which is a one-dimensional Abelian variety over , and is a principal homogeneous space over . If is not empty, then the choice of specifies an isomorphism under which becomes the trivial element of . In general, and are isomorphic over a finite extension of (see [1], [4], [13]).

Elliptic curves over the field of complex numbers.

An elliptic curve over is a compact Riemann surface of genus 1, and vice versa. The group structure turns into a complex Lie group, which is a one-dimensional complex torus , where is a lattice in the complex plane . Conversely, any one-dimensional complex torus is an elliptic curve (see [3]). From the topological point of view, an elliptic curve is a two-dimensional torus.

The theory of elliptic curves over is in essence equivalent to the theory of elliptic functions. An identification of a torus with an elliptic curve can be effected as follows. The elliptic functions with a given period lattice form a field generated by the Weierstrass -function (see Weierstrass elliptic functions) and its derivative , which are connected by the relation

The mapping () induces an isomorphism between the torus and the elliptic curve with equation . The identification of given by (1) with the torus is effected by curvilinear integrals of the holomorphic form and gives an isomorphism .

The description of the set of all elliptic curves as tori leads to the modular function . Two lattices and 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 is generated by the numbers 1 and in . Two lattices with bases and are similar if and only if for an element of the modular group . The modular function

is also called the absolute invariant; if and only if for some , and the function produces a one-to-one correspondence between the classes of isomorphic elliptic curves over and the complex numbers. If , then .

An elliptic curve has complex multiplication if and only if is an imaginary quadratic irrationality. In this case is a subring of finite index in the ring of algebraic integers of the imaginary quadratic field . Elliptic curves with complex multiplication are closely connected with the class field theory for imaginary quadratic fields (see [4], [8]).

Arithmetic of elliptic curves.

Let be an elliptic curve over the finite field with elements. The set is always non-empty and finite. Hence is endowed with the structure of a one-dimensional Abelian variety over , and with that of a finite Abelian group. The order of satisfies . The characteristic polynomial of the Frobenius endomorphism acting on the Tate module , , is . Its roots and are complex-conjugate algebraic integers of modulus . For any finite extension of of degree , the order of is . The zeta-function of is

For any algebraic integer of modulus in some imaginary quadratic field (or in ) one can find an elliptic curve over such that the order of is .

Let be the field of -adic numbers or a finite algebraic extension of it, let be the ring of integers of , let be an elliptic curve over , and suppose that is non-empty. The group structure turns into a commutative compact one-dimensional -adic Lie group (cf. Lie group, -adic). The group is Pontryagin-dual to the Weil–Châtelet group . If , then is a Tate curve (see [1], [5]) and there exists a canonical uniformization of analogous to the case of .

Let be an elliptic curve over for which is not empty. Then is biregularly isomorphic to the curve (1) with . Of all curves of the form (1) that are isomorphic to with integers and , one chooses the one for which the absolute value of the discriminant is minimal. The conductor and the -function of are defined as formal products of local factors:

(2)

over all prime numbers (see [1], [5], [13]). Here is some power of , and is a meromorphic function of the complex variable that has neither a zero nor a pole at . To determine the local factors one considers the reduction of modulo (), which is a plane projective curve over the residue class field and is given in an affine coordinate system by the equation

Let be the number of -points on . If does not divide , then is an elliptic curve over , and one puts

If divides , then the polynomial has a multiple root, and one puts

(depending on whether it is a triple or a double root). The product (2) converges in the right half-plane . It has been conjectured that has a meromorphic extension to the whole complex plane and that the function

(where is the gamma-function) satisfies the functional equation with (see [5], [3]). This conjecture has been proved for elliptic curves with complex multiplication.

The group is isomorphic to , where is a finite Abelian group and is a free Abelian group of a certain finite rank . is isomorphic to one of the following 15 groups (see [11]): , or , and , . The number is called the rank of the elliptic curve over , or its -rank. Examples are known of elliptic curves over of rank . There is a conjecture (see [1], [13]) that over there exist elliptic curves of arbitrary large rank.

In the study of one uses the Tate height , which is a non-negative definite quadratic form on (see [1], [3], [8], and also Height, in Diophantine geometry). For any the set is finite. In particular, vanishes precisely on the torsion subgroup of .

An important invariant of an elliptic curve is its Tate–Shafarevich group (see Weil–Châtelet group). The non-trivial elements of , an elliptic curve without -points, provide examples of elliptic curves for which the Hasse principle fails to hold. The group is periodic and for every the subgroup of its elements of order dividing is finite. For a large number of elliptic curves it has been verified that the 2- and -components of are finite (see [1], [4], [5]). There is a conjecture that is finite.

A conjecture of Birch and Swinnerton-Dyer asserts (see [5], [13]) that the order of the zero of the -function at is equal to the -rank of . In particular, has a zero at if and only if is infinite. So far (1984) the conjecture has not been proved for a single elliptic curve, but for elliptic curves with complex multiplication (and ) it has been established that when is infinite, then the -function has a zero at (see [14]). The conjecture of Birch and Swinnerton-Dyer gives the principal term of the asymptotic expansion of the -function as ; in it there occur the orders of the groups and and the determinant of the Tate height [1]. It can be restated in terms of the Tamagawa numbers (cf. Tamagawa number, see [7]).

There is a conjecture of Weil that an elliptic curve has a uniformization by modular functions relative to the congruence subgroup of the modular group (see [5] and also Zeta-function in algebraic geometry). This conjecture has been proved for elliptic functions with complex multiplication. It is known (see [15]) that every algebraic curve over can be uniformized (cf. Uniformization) by modular functions relative to some subgroup of of finite index.

References

[1] J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc. , 41 (1966) pp. 193–291
[2] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1968)
[3] D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974)
[4] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967)
[5] 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
[6] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91
[7] S. Bloch, "A note on height pairings, Tamagawa numbers, and the Birch and Swinnnerton-Dyer conjecture" Invent. Math. , 58 (1980) pp. 65–76
[8] S. Lang, "Elliptic curves; Diophantine analysis" , Springer (1978)
[9] S. Lang, "Elliptic functions" , Addison-Wesley (1973)
[10] B. Mazur, "Rational isogenies of prime degree" Invent. Math. , 44 (1978) pp. 129–162
[11] J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , Modular functions of one variable. 4 , Lect. notes in math. , 476 , Springer (1975)
[12] J.F. Mestre, "Construction d'une courbe elliptique de rang " C.R. Acad. Sci. Paris Sér. 1 , 295 (1982) pp. 643–644
[13] J. Tate, "The arithmetic of elliptic curves" Invent. Math. , 23 (1974) pp. 197–206
[14] J. Coates, A. Wiles, "On the conjecture of Birch and Swinnerton-Dyer" Invent. Math. , 39 (1977) pp. 223–251
[15] 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


Comments

References

[a1] B. Mazur, "Modular curves and the Eisenstein ideal" Publ. Math. IHES , 47 (1978) pp. 33–186
[a2] J.H. Silverman, "The arithmetic of elliptic curves" , Springer (1986)
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