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An [[Epimorphism|epimorphism]] of group schemes (cf. [[Group scheme|Group scheme]]) with a finite kernel. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i0527301.png" /> of group schemes over a ground scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i0527302.png" /> is said to be an isogeny if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i0527303.png" /> is surjective and if its kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i0527304.png" /> is a flat finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i0527305.png" />-scheme.
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In what follows it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i0527306.png" /> is the spectrum of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i0527307.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i0527308.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i0527309.png" /> is a group scheme of finite type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273010.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273011.png" /> be a finite subgroup scheme. Then the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273012.png" /> exists, and the natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273013.png" /> is an isogeny. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273014.png" /> is an isogeny of group schemes of finite type and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273016.png" />. For every isogeny <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273017.png" /> of Abelian varieties there exists an isogeny <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273018.png" /> such that the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273019.png" /> is the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273020.png" /> of multiplication of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273021.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273022.png" />. Composites of isogenies are isogenies. Two group schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273024.png" /> are said to be isogenous if there exists an isogeny <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273025.png" />. An isogeny <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273026.png" /> is said to be separable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273027.png" /> is an étale group scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273028.png" />. This is equivalent to the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273029.png" /> is a finite étale covering. An example of a separable isogeny is the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273032.png" /> is a finite field, then every separable isogeny <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273033.png" /> of connected commutative group schemes of dimension one factors through the isogeny <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273036.png" /> is the [[Frobenius endomorphism|Frobenius endomorphism]]. An example of a non-separable isogeny is the homomorphism of multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273037.png" /> in an Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273038.png" />.
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Localization of the additive category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273039.png" /> of Abelian varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273040.png" /> with respect to isogeny determines an Abelian category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273041.png" />, whose objects are called Abelian varieties up to isogeny. Every such object can be identified with an Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273042.png" />, and the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273044.png" /> are elements of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273045.png" /> over the field of rational numbers. An isogeny <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273046.png" /> defines an isomorphism of the corresponding objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273047.png" />. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273048.png" /> is semi-simple: each of its objects is isomorphic to a product of indecomposable objects. There is a complete description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273049.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273050.png" /> is a finite field (see [[#References|[4]]]).
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An [[Epimorphism|epimorphism]] of group schemes (cf. [[Group scheme|Group scheme]]) with a finite kernel. A morphism  $  f:  G \rightarrow G _ {1} $
 +
of group schemes over a ground scheme  $  S $
 +
is said to be an isogeny if  $  f $
 +
is surjective and if its kernel  $  \mathop{\rm Ker} ( f  ) $
 +
is a flat finite group  $  S $-
 +
scheme.
  
The concept of an isogeny is also defined for formal groups. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273051.png" /> of formal groups over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273052.png" /> is said to be an isogeny if its image in the quotient category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273053.png" /> of the category of formal groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273054.png" /> by the subcategory of Artinian formal groups is an isomorphism. An isogeny of group schemes determines an isogeny of the corresponding formal completions. There is a description of the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273055.png" /> of formal groups up to isogeny (see [[#References|[1]]], [[#References|[2]]]).
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In what follows it is assumed that  $  S $
 +
is the spectrum of a field  $  k $
 +
of characteristic  $  p \geq  0 $.
 +
Suppose that  $  G $
 +
is a group scheme of finite type over  $  k $,
 +
and let  $  H $
 +
be a finite subgroup scheme. Then the quotient  $  G/H $
 +
exists, and the natural mapping  $  G \rightarrow G/H $
 +
is an isogeny. Conversely, if  $  f:  G \rightarrow G _ {1} $
 +
is an isogeny of group schemes of finite type and  $  H =  \mathop{\rm ker} ( f  ) $,
 +
then  $  G _ {1} = G/H $.
 +
For every isogeny  $  f:  G \rightarrow G _ {1} $
 +
of Abelian varieties there exists an isogeny  $  g:  G _ {1} \rightarrow G $
 +
such that the composite  $  g \circ f $
 +
is the homomorphism  $  n _ {G} $
 +
of multiplication of  $  G $
 +
by  $  n $.
 +
Composites of isogenies are isogenies. Two group schemes  $  G $
 +
and  $  G _ {1} $
 +
are said to be isogenous if there exists an isogeny  $  f:  G \rightarrow G _ {1} $.
 +
An isogeny  $  f:  G \rightarrow G _ {1} $
 +
is said to be separable if  $  \mathop{\rm ker} ( f  ) $
 +
is an étale group scheme over  $  k $.
 +
This is equivalent to the fact that  $  f $
 +
is a finite étale covering. An example of a separable isogeny is the homomorphism  $  n _ {G} $,
 +
where  $  ( n, p) = 1 $.
 +
If  $  k $
 +
is a finite field, then every separable isogeny  $  f:  G \rightarrow G _ {1} $
 +
of connected commutative group schemes of dimension one factors through the isogeny  $  \mathfrak p:  G \rightarrow G $,
 +
where  $  \mathfrak p = F -  \mathop{\rm id} _ {G} $
 +
and  $  F $
 +
is the [[Frobenius endomorphism|Frobenius endomorphism]]. An example of a non-separable isogeny is the homomorphism of multiplication by  $  n = p  ^ {r} $
 +
in an Abelian variety  $  A $.
 +
 
 +
Localization of the additive category  $  A ( k) $
 +
of Abelian varieties over  $  k $
 +
with respect to isogeny determines an Abelian category  $  M ( k) $,
 +
whose objects are called Abelian varieties up to isogeny. Every such object can be identified with an Abelian variety  $  A $,
 +
and the morphisms  $  A \rightarrow A _ {1} $
 +
in  $  M ( k) $
 +
are elements of the algebra  $  \mathop{\rm Hom} _ {A ( k) }  ( A, A _ {1} ) \otimes _ {\mathbf Z }  \mathbf Q $
 +
over the field of rational numbers. An isogeny  $  f:  A \rightarrow A _ {1} $
 +
defines an isomorphism of the corresponding objects in  $  M ( k) $.
 +
The category  $  M ( k) $
 +
is semi-simple: each of its objects is isomorphic to a product of indecomposable objects. There is a complete description of  $  M ( k) $
 +
when  $  k $
 +
is a finite field (see [[#References|[4]]]).
 +
 
 +
The concept of an isogeny is also defined for formal groups. A morphism $  f: G \rightarrow G _ {1} $
 +
of formal groups over a field $  k $
 +
is said to be an isogeny if its image in the quotient category $  \phi ( k) $
 +
of the category of formal groups over $  k $
 +
by the subcategory of Artinian formal groups is an isomorphism. An isogeny of group schemes determines an isogeny of the corresponding formal completions. There is a description of the category $  \phi ( k) $
 +
of formal groups up to isogeny (see [[#References|[1]]], [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,  "The theory of commutative formal groups over fields of finite characteristic"  ''Russian Math. Surveys'' , '''18''' :  6  (1963)  pp. 1–81  ''Uspekhi Mat. Nauk'' , '''18''' :  6  (1963)  pp. 3–90</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Mumford,  "Abelian varieties" , Oxford Univ. Press  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Groupes algébrique et corps des classes" , Hermann  (1959)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.T. Tate,  "Classes d'isogénie des variétés abéliennes sur un corps fini (d' après T. Honda)" , ''Sem. Bourbaki Exp. 352'' , ''Lect. notes in math.'' , '''179''' , Springer  (1968/69)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Dieudonné,  "Groupes de Lie et hyperalgèbres de Lie sur un corps de characteristique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273056.png" />"  ''Comm. Math. Helvetici'' , '''28''' :  1  (1954)  pp. 87–118</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,  "The theory of commutative formal groups over fields of finite characteristic"  ''Russian Math. Surveys'' , '''18''' :  6  (1963)  pp. 1–81  ''Uspekhi Mat. Nauk'' , '''18''' :  6  (1963)  pp. 3–90</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Mumford,  "Abelian varieties" , Oxford Univ. Press  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Groupes algébrique et corps des classes" , Hermann  (1959)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.T. Tate,  "Classes d'isogénie des variétés abéliennes sur un corps fini (d' après T. Honda)" , ''Sem. Bourbaki Exp. 352'' , ''Lect. notes in math.'' , '''179''' , Springer  (1968/69)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Dieudonné,  "Groupes de Lie et hyperalgèbres de Lie sur un corps de characteristique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052730/i05273056.png" />"  ''Comm. Math. Helvetici'' , '''28''' :  1  (1954)  pp. 87–118</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Honda,  "Isogeny classes of Abelian varieties over finite fields"  ''Math. Soc. Japan'' , '''20'''  (1968)  pp. 83–95</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Tate,  "Endomorphisms of Abelian varieties over finite fields"  ''Invent. Math.'' , '''2'''  (1966)  pp. 134–144</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Honda,  "Isogeny classes of Abelian varieties over finite fields"  ''Math. Soc. Japan'' , '''20'''  (1968)  pp. 83–95</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Tate,  "Endomorphisms of Abelian varieties over finite fields"  ''Invent. Math.'' , '''2'''  (1966)  pp. 134–144</TD></TR></table>

Revision as of 22:13, 5 June 2020


An epimorphism of group schemes (cf. Group scheme) with a finite kernel. A morphism $ f: G \rightarrow G _ {1} $ of group schemes over a ground scheme $ S $ is said to be an isogeny if $ f $ is surjective and if its kernel $ \mathop{\rm Ker} ( f ) $ is a flat finite group $ S $- scheme.

In what follows it is assumed that $ S $ is the spectrum of a field $ k $ of characteristic $ p \geq 0 $. Suppose that $ G $ is a group scheme of finite type over $ k $, and let $ H $ be a finite subgroup scheme. Then the quotient $ G/H $ exists, and the natural mapping $ G \rightarrow G/H $ is an isogeny. Conversely, if $ f: G \rightarrow G _ {1} $ is an isogeny of group schemes of finite type and $ H = \mathop{\rm ker} ( f ) $, then $ G _ {1} = G/H $. For every isogeny $ f: G \rightarrow G _ {1} $ of Abelian varieties there exists an isogeny $ g: G _ {1} \rightarrow G $ such that the composite $ g \circ f $ is the homomorphism $ n _ {G} $ of multiplication of $ G $ by $ n $. Composites of isogenies are isogenies. Two group schemes $ G $ and $ G _ {1} $ are said to be isogenous if there exists an isogeny $ f: G \rightarrow G _ {1} $. An isogeny $ f: G \rightarrow G _ {1} $ is said to be separable if $ \mathop{\rm ker} ( f ) $ is an étale group scheme over $ k $. This is equivalent to the fact that $ f $ is a finite étale covering. An example of a separable isogeny is the homomorphism $ n _ {G} $, where $ ( n, p) = 1 $. If $ k $ is a finite field, then every separable isogeny $ f: G \rightarrow G _ {1} $ of connected commutative group schemes of dimension one factors through the isogeny $ \mathfrak p: G \rightarrow G $, where $ \mathfrak p = F - \mathop{\rm id} _ {G} $ and $ F $ is the Frobenius endomorphism. An example of a non-separable isogeny is the homomorphism of multiplication by $ n = p ^ {r} $ in an Abelian variety $ A $.

Localization of the additive category $ A ( k) $ of Abelian varieties over $ k $ with respect to isogeny determines an Abelian category $ M ( k) $, whose objects are called Abelian varieties up to isogeny. Every such object can be identified with an Abelian variety $ A $, and the morphisms $ A \rightarrow A _ {1} $ in $ M ( k) $ are elements of the algebra $ \mathop{\rm Hom} _ {A ( k) } ( A, A _ {1} ) \otimes _ {\mathbf Z } \mathbf Q $ over the field of rational numbers. An isogeny $ f: A \rightarrow A _ {1} $ defines an isomorphism of the corresponding objects in $ M ( k) $. The category $ M ( k) $ is semi-simple: each of its objects is isomorphic to a product of indecomposable objects. There is a complete description of $ M ( k) $ when $ k $ is a finite field (see [4]).

The concept of an isogeny is also defined for formal groups. A morphism $ f: G \rightarrow G _ {1} $ of formal groups over a field $ k $ is said to be an isogeny if its image in the quotient category $ \phi ( k) $ of the category of formal groups over $ k $ by the subcategory of Artinian formal groups is an isomorphism. An isogeny of group schemes determines an isogeny of the corresponding formal completions. There is a description of the category $ \phi ( k) $ of formal groups up to isogeny (see [1], [2]).

References

[1] Yu.I. Manin, "The theory of commutative formal groups over fields of finite characteristic" Russian Math. Surveys , 18 : 6 (1963) pp. 1–81 Uspekhi Mat. Nauk , 18 : 6 (1963) pp. 3–90
[2] D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974)
[3] J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959)
[4] J.T. Tate, "Classes d'isogénie des variétés abéliennes sur un corps fini (d' après T. Honda)" , Sem. Bourbaki Exp. 352 , Lect. notes in math. , 179 , Springer (1968/69)
[5] J. Dieudonné, "Groupes de Lie et hyperalgèbres de Lie sur un corps de characteristique " Comm. Math. Helvetici , 28 : 1 (1954) pp. 87–118

Comments

References

[a1] T. Honda, "Isogeny classes of Abelian varieties over finite fields" Math. Soc. Japan , 20 (1968) pp. 83–95
[a2] J. Tate, "Endomorphisms of Abelian varieties over finite fields" Invent. Math. , 2 (1966) pp. 134–144
How to Cite This Entry:
Isogeny. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isogeny&oldid=12388
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article