In mathematics, in particular, in algebraic geometry, an isogeny is a
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
of
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
s (also known as group varieties) that is
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
and has a finite
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
.
If the
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
are
abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...
, then any morphism of the underlying algebraic varieties which is surjective with finite
fibres
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
is automatically an isogeny, provided that . Such an isogeny then provides a
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
wh ...
between the groups of -valued points of and , for any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
over which is defined.
The terms "isogeny" and "isogenous" come from the Greek word ισογενη-ς, meaning "equal in kind or nature". The term "isogeny" was introduced by
Weil; before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny.
Case of abelian varieties
For
abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...
, such as
elliptic curves
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
, this notion can also be formulated as follows:
Let ''E''
1 and ''E''
2 be abelian varieties of the same dimension over a field ''k''. An isogeny between ''E''
1 and ''E''
2 is a dense morphism of varieties that preserves basepoints (i.e. ''f'' maps the identity point on ''E''
1 to that on ''E''
2).
This is equivalent to the above notion, as every dense morphism between two abelian varieties of the same dimension is automatically surjective with finite fibres, and if it preserves identities then it is a homomorphism of groups.
Two abelian varieties ''E''
1 and ''E''
2 are called isogenous if there is an isogeny . This can be shown to be an equivalence relation; in the case of elliptic curves, symmetry is due to the existence of the
dual isogeny In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''K''.
Definition
To an abelian variety ''A'' over a field ''k'', one associates a dual abelian variety ''A''v (over the same field), which ...
. As above, every isogeny induces homomorphisms of the groups of the k-valued points of the abelian varieties.
See also
*
Abelian varieties up to isogeny In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in gene ...
*
Selmer group
In arithmetic geometry, the Selmer group, named in honor of the work of by , is a group constructed from an isogeny of abelian varieties.
The Selmer group of an isogeny
The Selmer group of an abelian variety ''A'' with respect to an isogeny ''f ...
References
*
* {{cite book , last=Mumford , first=David , author-link=David Mumford , year=1974 , title=Abelian Varieties , publisher=Oxford University Press , isbn=0-19-560528-4
Morphisms of schemes