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A non-singular complete
+
{{MSC|14h57|11Gxx,14K15}}
 +
{{TEX|done}}
 +
 
 +
 
 +
An ''elliptic curve'' is a non-singular complete
 
[[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic
 
[[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic
 
curves is the source of a large part of contemporary algebraic
 
curves is the source of a large part of contemporary algebraic
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over an algebraically closed field $k$. Then $X$ is biregularly
 
over an algebraically closed field $k$. Then $X$ is biregularly
 
isomorphic to a plane cubic curve (see
 
isomorphic to a plane cubic curve (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[9]]],
+
{{Cite|La2}},
[[#References|[13]]]). If $\mathop{\mathrm{char}} k \ne 2,3$, then in the projective plane ${\mathbb P}^2$ there
+
{{Cite|Ta}}). If $\mathop{\mathrm{char}} k \ne 2,3$, then in the projective plane ${\mathbb P}^2$ there
 
is an affine coordinate system in which the equation of $X$ is in
 
is an affine coordinate system in which the equation of $X$ is in
 
normal Weierstrass form:  
 
normal Weierstrass form:  
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varieties $g:(Y,Q_0)\to(X,P_0)$ such that $gh=n_X$ and $hg=n_Y$  
 
varieties $g:(Y,Q_0)\to(X,P_0)$ such that $gh=n_X$ and $hg=n_Y$  
 
for some $n$ (see
 
for some $n$ (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[6]]]).
+
{{Cite|Ha}}).
  
 
The automorphism group of an elliptic curve $X$ acts transitively on
 
The automorphism group of an elliptic curve $X$ acts transitively on
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is neither 0 nor $1728$, then $G$ consists of the two elements $1_X$ and
 
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
 
$(-1)_X$. The order of $G$ is 4 when $j(X)=1728$ and 6 when $j(X)=0$ (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[6]]],
+
{{Cite|Ha}},
[[#References|[13]]]).
+
{{Cite|Ta}}).
  
 
An important invariant of an elliptic curve is the endomorphism ring
 
An important invariant of an elliptic curve is the endomorphism ring
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complex multiplication. The ring $R$ can be of one of the following
 
complex multiplication. The ring $R$ can be of one of the following
 
types (see
 
types (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[9]]],
+
{{Cite|La2}},
[[#References|[13]]]): i) $R={\mathbb Z}$; ii) $R={\mathbb Z}+f{\mathcal O}\subset {\mathcal O}$, where $\mathcal O$ is the ring of
+
{{Cite|Ta}}): i) $R={\mathbb Z}$; ii) $R={\mathbb Z}+f{\mathcal O}\subset {\mathcal O}$, where $\mathcal O$ is the ring of
 
algebraic integers of an imaginary quadratic field $k$ and $f\in {\mathbb N}$; or
 
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
 
iii) $R$ is a non-commutative ${\mathbb Z}$-algebra of rank 4 without divisors
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general, $X$ and $J(X)$ are isomorphic over a finite extension of $k$
 
general, $X$ and $J(X)$ are isomorphic over a finite extension of $k$
 
(see
 
(see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[4]]],
+
{{Cite|CaFr}},
[[#References|[13]]]).
+
{{Cite|Ta}}).
  
 
==Elliptic curves over the field of complex numbers.==
 
==Elliptic curves over the field of complex numbers.==
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complex plane ${\mathbb C}$. Conversely, any one-dimensional complex torus is an
 
complex plane ${\mathbb C}$. Conversely, any one-dimensional complex torus is an
 
elliptic curve (see
 
elliptic curve (see
[[#References|[3]]]). From the topological point of view, an elliptic
+
{{Cite|Mu}}). From the topological point of view, an elliptic
 
curve is a two-dimensional torus.
 
curve is a two-dimensional torus.
  
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[[Weierstrass elliptic functions|Weierstrass elliptic functions]]) and
 
[[Weierstrass elliptic functions|Weierstrass elliptic functions]]) and
 
its derivative $\wp'(z)$, which are connected by the relation  
 
its derivative $\wp'(z)$, which are connected by the relation  
$$\wp'=4\wp^3 - g_2 - g_3$$
+
$$\wp'=4\wp^3 - g_2\wp - g_3$$
 
The
 
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
 
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
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[[Class field theory|class field theory]] for imaginary quadratic
 
[[Class field theory|class field theory]] for imaginary quadratic
 
fields (see
 
fields (see
[[#References|[4]]],
+
{{Cite|CaFr}},
[[#References|[8]]]).
+
{{Cite|La}}).
  
 
==Arithmetic of elliptic curves.==
 
==Arithmetic of elliptic curves.==
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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
 
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 $k$
 
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
+
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  
 
[[Zeta-function|zeta-function]] of $X$ is  
 
$$\frac{(1-q^{-s})(1-q^{1-s})}{1-(q+1-A)q^{-s}+q^{1-2s}}.$$
 
$$\frac{(1-q^{-s})(1-q^{1-s})}{1-(q+1-A)q^{-s}+q^{1-2s}}.$$
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structure turns $X(k)$ into a commutative compact one-dimensional
 
structure turns $X(k)$ into a commutative compact one-dimensional
 
$p$-adic Lie group (cf.
 
$p$-adic Lie group (cf.
[[Lie-group-adic|Lie group, $p$-adic]]). The group $X(k)$ is
+
[[Lie-group, p-adic|Lie group, $p$-adic]]). The group $X(k)$ is
 
Pontryagin-dual to the
 
Pontryagin-dual to the
 
[[Weil–Châtelet group|Weil–Châtelet group]] ${\rm WC}(k,X)$. If $j(X)\notin B$, then $X$ is a
 
[[Weil–Châtelet group|Weil–Châtelet group]] ${\rm WC}(k,X)$. If $j(X)\notin B$, then $X$ is a
 
Tate curve (see
 
Tate curve (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[5]]]) and there exists a canonical uniformization of
+
{{Cite|Ma}}) and there exists a canonical uniformization of
 
$X(k)$ analogous to the case of ${\mathbb C}$.
 
$X(k)$ analogous to the case of ${\mathbb C}$.
  
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over all prime
 
over all prime
 
numbers $p$ (see
 
numbers $p$ (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[5]]],
+
{{Cite|Ma}},
[[#References|[13]]]). Here $f_p$ is some power of $p$, and
+
{{Cite|Ta}}). Here $f_p$ is some power of $p$, and
 
$L_p(X,s)$ is a
 
$L_p(X,s)$ is a
 
meromorphic function of the complex variable $s$ that has neither a
 
meromorphic function of the complex variable $s$ that has neither a
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[[Gamma-function|gamma-function]]) satisfies the functional equation
 
[[Gamma-function|gamma-function]]) satisfies the functional equation
 
$\xi_X(s) = W\xi_X(2-s)$ with $W=\pm1$ (see
 
$\xi_X(s) = W\xi_X(2-s)$ with $W=\pm1$ (see
[[#References|[5]]],
+
{{Cite|Ma}},
[[#References|[3]]]). This conjecture has been proved for elliptic
+
{{Cite|Mu}}). This conjecture has been proved for elliptic
 
curves with complex multiplication.
 
curves with complex multiplication.
  
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group and $F$ is a free Abelian group of a certain finite rank
 
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
 
$r$. $X({\mathbb Q})_t$ is isomorphic to one of the following 15 groups (see
[[#References|[11]]]): ${\mathbb Z}/m{\mathbb Z}$, $1\le m\le 10$ or $m = 12$, and $({\mathbb Z}/2{\mathbb Z})\times ({\mathbb Z}/\nu{\mathbb Z})$, $1\le \nu\le 4$. The number $r$
+
{{Cite|SeDeKu}}): ${\mathbb Z}/m{\mathbb Z}$, $1\le m\le 10$ or $m = 12$, and $({\mathbb Z}/2{\mathbb Z})\times ({\mathbb Z}/\nu{\mathbb Z})$, $1\le \nu\le 4$. The number $r$
 
is called the rank of the elliptic curve over ${\mathbb Q}$, or its
 
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
 
${\mathbb Q}$-rank. Examples are known of elliptic curves over ${\mathbb Q}$ of rank
 
$\ge 12$. There is a conjecture (see
 
$\ge 12$. There is a conjecture (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[13]]]) that over ${\mathbb Q}$ there exist elliptic curves of
+
{{Cite|Ta}}) that over ${\mathbb Q}$ there exist elliptic curves of
 
arbitrary large rank.
 
arbitrary large rank.
  
 
In the study of $X({\mathbb Q})$ one uses the Tate height ${\hat h}:X({\mathbb Q})\to {\mathbb R}^+$, which is a
 
In the study of $X({\mathbb Q})$ one uses the Tate height ${\hat h}:X({\mathbb Q})\to {\mathbb R}^+$, which is a
 
non-negative definite quadratic form on $X({\mathbb Q})$ (see
 
non-negative definite quadratic form on $X({\mathbb Q})$ (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[3]]],
+
{{Cite|Mu}},
[[#References|[8]]], and also
+
{{Cite|La}}, and also
 
[[Height, in Diophantine geometry|Height, in Diophantine
 
[[Height, in Diophantine geometry|Height, in Diophantine
 
geometry]]). For any $c\in {\mathbb R}^+$ the set $\{P\in X(\mathbb Q) | {\hat h}(P)\le c\}$ is finite. In particular, $\hat h$
 
geometry]]). For any $c\in {\mathbb R}^+$ the set $\{P\in X(\mathbb Q) | {\hat h}(P)\le c\}$ is finite. In particular, $\hat h$
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dividing $n$ is finite. For a large number of elliptic curves it has
 
dividing $n$ is finite. For a large number of elliptic curves it has
 
been verified that the 2- and $3$-components of ${\rm Sha}$ are finite (see
 
been verified that the 2- and $3$-components of ${\rm Sha}$ are finite (see
[[#References|[1]]],
+
{{Cite|Ca}},
[[#References|[4]]],
+
{{Cite|CaFr}},
[[#References|[5]]]). There is a conjecture that ${\rm Sha}$ is finite.
+
{{Cite|Ma}}). There is a conjecture that ${\rm Sha}$ is finite.
  
 
A conjecture of Birch and Swinnerton-Dyer asserts (see
 
A conjecture of Birch and Swinnerton-Dyer asserts (see
[[#References|[5]]],
+
{{Cite|Ma}},
[[#References|[13]]]) that the order of the zero of the $L$-function
+
{{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
 
$L(X,s)$ 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
 
zero at $s=1$ if and only if $X({\mathbb Q})$ is infinite. So far (1984) the
Line 284: Line 288:
 
established that when $X({\mathbb Q})$ is infinite, then the $L$-function has a
 
established that when $X({\mathbb Q})$ is infinite, then the $L$-function has a
 
zero at $s=1$ (see
 
zero at $s=1$ (see
[[#References|[14]]]). The conjecture of Birch and Swinnerton-Dyer
+
{{Cite|CoWi}}). The conjecture of Birch and Swinnerton-Dyer
 
gives the principal term of the asymptotic expansion of the
 
gives the principal term of the asymptotic expansion of the
 
$L$-function as $s\to 1$; in it there occur the orders of the groups ${\rm Sha}\;(X)$
 
$L$-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
 
and $X({\mathbb Q})_t$ and the determinant of the Tate height
[[#References|[1]]]. It can be restated in terms of the Tamagawa
+
{{Cite|Ca}}. It can be restated in terms of the Tamagawa
 
numbers (cf.
 
numbers (cf.
 
[[Tamagawa number|Tamagawa number]], see
 
[[Tamagawa number|Tamagawa number]], see
[[#References|[7]]]).
+
{{Cite|Bl}}).
  
 
There is a conjecture of Weil that an elliptic curve $X$ has a
 
There is a conjecture of Weil that an elliptic curve $X$ has a
 
uniformization by modular functions relative to the congruence
 
uniformization by modular functions relative to the congruence
 
subgroup $\Gamma_0(N)$ of the modular group $\Gamma$ (see
 
subgroup $\Gamma_0(N)$ of the modular group $\Gamma$ (see
[[#References|[5]]] and also
+
{{Cite|Ma}} and also
 
[[Zeta-function|Zeta-function]] in algebraic geometry). This
 
[[Zeta-function|Zeta-function]] in algebraic geometry). This
 
conjecture has been proved for elliptic functions with complex
 
conjecture has been proved for elliptic functions with complex
 
multiplication. It is known (see
 
multiplication. It is known (see
[[#References|[15]]]) that every algebraic curve over $\mathbb Q$ can be
+
{{Cite|Be}}) that every algebraic curve over $\mathbb Q$ can be
 
uniformized (cf.
 
uniformized (cf.
 
[[Uniformization|Uniformization]]) by modular functions relative to
 
[[Uniformization|Uniformization]]) by modular functions relative to
Line 306: Line 310:
  
 
====References====
 
====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'''
+
|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}}         
(1966) pp. 193–291</TD></TR><TR><TD valign="top">[2]</TD> <TD
+
|-
valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine
+
|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}}     
Funktionentheorie und elliptische Funktionen" , Springer
+
|-
(1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
+
|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}}         
D. Mumford, "Abelian varieties" , Oxford Univ. Press
+
|-
(1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
+
|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}}       
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"|{{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"> Yu.I. Manin, "Cyclotomic fields and modular curves"
+
|-
''Russian Math. Surveys'' , '''26''' : 6 (1971) pp. 6–78 ''Uspekhi
+
|valign="top"|{{Ref|Ha}}||valign="top"| R. Hartshorne, "Algebraic geometry", Springer (1977) pp. 91 {{MR|0463157}}  {{ZBL|0367.14001}}         
Mat. Nauk'' , '''26''' : 6 (1971) pp. 7–71</TD></TR><TR><TD
+
|-
valign="top">[6]</TD> <TD valign="top"> R. Hartshorne, "Algebraic
+
|valign="top"|{{Ref|HuCo}}||valign="top"| A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen", Springer (1964) {{MR|0173749}}  {{ZBL|0135.12101}}
geometry" , Springer (1977) pp. 91</TD></TR><TR><TD
+
|-
valign="top">[7]</TD> <TD valign="top"> S. Bloch, "A note on height
+
|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Elliptic curves: Diophantine analysis", Springer (1978)   {{MR|0518817}}  {{ZBL|0388.10001}}   
pairings, Tamagawa numbers, and the Birch and Swinnnerton-Dyer
+
|-
conjecture" ''Invent. Math.'' , '''58''' (1980)
+
|valign="top"|{{Ref|La2}}||valign="top"| S. Lang, "Elliptic functions", Addison-Wesley (1973) {{MR|0409362}}  {{ZBL|0316.14001}}         
pp. 65–76</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">
+
|-
S. Lang, "Elliptic curves; Diophantine analysis" , Springer
+
|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}}         
(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"|{{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">[10]</TD> <TD valign="top"> B. Mazur, "Rational isogenies
+
|-
of prime degree" ''Invent. Math.'' , '''44''' (1978)
+
|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}}
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
+
|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}}         
functions of one variable. 4'' , ''Lect. notes in math.'' , '''476'''
+
|-
, Springer (1975)</TD></TR><TR><TD valign="top">[12]</TD> <TD
+
|valign="top"|{{Ref|Mu}}||valign="top"| D. Mumford, "Abelian varieties", Oxford Univ. Press (1974)   {{ZBL|0326.14012}}         
valign="top"> J.F. Mestre, "Construction d'une courbe elliptique de
+
|-
rang $\ge 12$" ''C.R. Acad. Sci. Paris Sér. 1'' , '''295''' (1982)
+
|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}}         
pp. 643–644</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">
+
|-
J. Tate, "The arithmetic of elliptic curves" ''Invent. Math.'' ,
+
|valign="top"|{{Ref|Si}}||valign="top"| J.H. Silverman, "The arithmetic of elliptic curves", Springer (1986) {{MR|0817210}}  {{ZBL|0585.14026}}         
'''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
+
|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}}         
Swinnerton-Dyer" ''Invent. Math.'' , '''39''' (1977)
+
|-
pp. 223–251</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">
+
|}
G.V. Belyi, "On Galois extensions of a maximal cyclotomic field"
 
''Math. USSR Izv.'' , '''14''' : 2 (1980) pp. 247–256
 
''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''43''' (1979)
 
pp. 267–276</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
 
 
 
 
====References====
 
<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>
 

Latest revision as of 20:37, 19 September 2017

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


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

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

The geometry of an elliptic curve.

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

The group structure on an elliptic curve.

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

An elliptic curve as a one-dimensional Abelian variety.

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

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

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

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

Elliptic curves over non-closed fields.

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

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

Elliptic curves over the field of complex numbers.

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

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

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

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

Arithmetic of elliptic curves.

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

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

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

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

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

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

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

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

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

References

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How to Cite This Entry:
Elliptic curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_curve&oldid=19588
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