Isogeny

<h2 id="degree-of-isogeny">Degree of isogeny</h2>
Let f : A → B be isogeny between two algebraic groups.
This mapping induces a pullback mapping f* : K(B) → K(A) between their <a href="/facts/Function_field_of_an_algebraic_variety/54cc62wP">rational function fields</a>. Since the mapping is nontrivial, it is a field embedding and 
 
 
 
 im
 ⁡
 
 f
 
 ∗
 
 
 
 
 {\displaystyle \operatorname {im} f^{*}}
 
 is a subfield of K(A). The <a href="/facts/Degree_of_a_field_extension/hY5LgHYk">degree</a> of the extension 
 
 
 
 K
 (
 A
 )
 
 /
 
 im
 ⁡
 
 f
 
 ∗
 
 
 
 
 {\displaystyle K(A)/\operatorname {im} f^{*}}
 
 is called degree of isogeny:

deg
        ⁡
        f
        :=
        [
        K
        (
        A
        )
        :
        im
        ⁡
        
          f
          
            ∗
          
        
        ]
      
    
    {\displaystyle \deg f:=[K(A):\operatorname {im} f^{*}]}

Properties of degree:

<ul><li>If 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\rightarrow Y}
 
, 
 
 
 
 g
 :
 Y
 →
 Z
 
 
 {\displaystyle g:Y\rightarrow Z}
 
 are isogenies of algebraic groups, then: 
 
 
 
 deg
 ⁡
 (
 g
 ∘
 f
 )
 =
 deg
 ⁡
 g
 ⋅
 deg
 ⁡
 f
 
 
 {\displaystyle \deg(g\circ f)=\deg g\cdot \deg f}
 
</li>
<li>If 
 
 
 
 c
 h
 a
 r
 
 K
 ∤
 deg
 ⁡
 f
 
 
 {\displaystyle char\;K\nmid \deg f}
 
, then 
 
 
 
 deg
 ⁡
 f
 =
 
 |
 
 ker
 
 f
 
 |
 
 
 
 {\displaystyle \deg f=|\ker \;f|}
 
</li></ul>
<h2 id="case-of-abelian-varieties">Case of abelian varieties</h2>

For <a href="/facts/Abelian_varieties/dGSDVruG">abelian varieties</a>, such as <a href="/facts/Elliptic_curves/60jvaHSA">elliptic curves</a>, this notion can also be formulated as follows:
Let E1 and E2 be abelian varieties of the same dimension over a field k. An isogeny between E1 and E2 is a dense morphism f : E1 → E2 of varieties that preserves basepoints (i.e. f maps the identity point on E1 to that on E2).
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 E1 and E2 are called isogenous if there is an isogeny E1 → E2. This can be shown to be an equivalence relation; in the case of elliptic curves, symmetry is due to the existence of the <a href="/facts/Dual_isogeny/xRSRm5KU">dual isogeny</a>. As above, every isogeny induces homomorphisms of the groups of the k-valued points of the abelian varieties.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Abelian_varieties_up_to_isogeny/NJ36V2lu">Abelian varieties up to isogeny</a></li>
<li><a href="/facts/Selmer_group/jnT5RIqE">Selmer group</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1983). Abelian Varieties. Springer Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-90875-7.</li>
<li><a href="/facts/David_Mumford/WJoJ8K2u">Mumford, David</a> (1974). Abelian Varieties. Oxford University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-560528-4.</li></ul>

Isogeny open-in-new

Isogeny