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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.

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====
<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====

====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>