# Isogeny

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{{Short description|Type of map between algebraic groups}}
{{no footnotes|date=June 2024}}
In mathematics, particularly in [algebraic geometry](/source/algebraic_geometry), an '''isogeny''' is a [morphism](/source/Morphism_of_algebraic_varieties) of [algebraic group](/source/algebraic_group)s (also known as group varieties) that is [surjective](/source/surjective_function) and has a finite [kernel](/source/kernel_(algebra)).

If the [groups](/source/group_(mathematics)) are [abelian varieties](/source/abelian_variety), then any morphism {{math|''f'' : ''A'' → ''B''}} of the underlying algebraic varieties which is surjective with finite [fibres](/source/fiber_(mathematics)) is automatically an isogeny, provided that {{math|1=''f''(1{{sub|''A''}}) = 1{{sub|''B''}}}}. Such an isogeny {{math|''f''}} then provides a [group homomorphism](/source/group_homomorphism) between the groups of {{math|''k''}}-valued points of {{math|''A''}} and {{math|''B''}}, for any [field](/source/field_(mathematics)) {{math|''k''}} over which {{math|''f''}} 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](/source/Andr%C3%A9_Weil); before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny.

==Degree of isogeny==
Let {{math|''f'' : ''A'' → ''B''}} be isogeny between two algebraic groups.
This mapping induces a pullback mapping {{math|''f*''&nbsp;:&nbsp;''K(B)''&nbsp;→&nbsp;''K(A)''}} between their [rational function fields](/source/Function_field_of_an_algebraic_variety). Since the mapping is nontrivial, it is a field embedding and <math>\operatorname{im} f^*</math> is a subfield of {{math|''K(A)''}}. The [degree](/source/degree_of_a_field_extension) of the extension <math>K(A) / \operatorname{im} f^* </math> is called degree of isogeny:
:<math> \deg f := [K(A): \operatorname{im} f^*]</math>

Properties of degree:
* If <math>f:X \rightarrow Y</math>, <math>g:Y \rightarrow Z</math> are isogenies of algebraic groups, then: <math>\deg (g\circ f)=\deg g \cdot \deg f </math>
* If <math>char \; K \nmid \deg f</math>, then <math>\deg f = |\ker\; f|</math>

==Case of abelian varieties==
right|thumb|300px|Isogenous elliptic curves to ''E'' can be obtained by quotienting ''E'' by finite subgroups, here subgroups of the 4-torsion subgroup.

For [abelian varieties](/source/abelian_varieties), such as [elliptic curves](/source/elliptic_curves), this notion can also be formulated as follows:

Let ''E''<sub>1</sub> and ''E''<sub>2</sub> be abelian varieties of the same dimension over a field ''k''. An '''isogeny''' between ''E''<sub>1</sub> and ''E''<sub>2</sub> is a dense morphism {{math|''f''&nbsp;:&nbsp;''E''<sub>1</sub>&nbsp;→&nbsp;''E''<sub>2</sub>}} of varieties that preserves basepoints (i.e. ''f'' maps the identity point on ''E''<sub>1</sub> to that on ''E''<sub>2</sub>).

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''<sub>1</sub> and ''E''<sub>2</sub> are called '''isogenous''' if there is an isogeny {{math|''E''<sub>1</sub>&nbsp;→&nbsp;''E''<sub>2</sub>}}. 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](/source/dual_isogeny). 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](/source/Abelian_varieties_up_to_isogeny)
* [Selmer group](/source/Selmer_group)

==References==
{{reflist}}

== External links ==
* {{cite book |last=Lang |first=Serge |author-link=Serge Lang |year=1983 |title=Abelian Varieties |publisher=Springer Verlag |isbn=3-540-90875-7}}
* {{cite book |last=Mumford |first=David |author-link=David Mumford |year=1974 |title=Abelian Varieties |publisher=Oxford University Press |isbn=0-19-560528-4}}

Category:Morphisms of schemes

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Adapted from the Wikipedia article [Isogeny](https://en.wikipedia.org/wiki/Isogeny) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Isogeny?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.