Isogeny

Let f : A → B be isogeny between two algebraic groups. This mapping induces a pullback mapping f* : K(B) → K(A) between their rational function fields. Since mapping is nontrivial, it is a field embedding and ${\displaystyle \operatorname {im} f^{*}}$  is a subfield of K(A). The degree of the extension ${\displaystyle K(A)/\operatorname {im} f^{*}}$  is called degree of isogeny:

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

Properties of degree:

• If ${\displaystyle f:X\rightarrow Y}$ , ${\displaystyle g:Y\rightarrow Z}$  are isogenies of algebraic groups, then: ${\displaystyle \deg(g\circ f)=\deg g\cdot \deg f}$ 
• If ${\displaystyle char\;K\nmid \deg f}$ , then ${\displaystyle \deg f=|\ker \;f|}$ 

 

For abelian varieties, such as elliptic curves, 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 dual isogeny. As above, every isogeny induces homomorphisms of the groups of the k-valued points of the abelian varieties.

• Abelian varieties up to isogeny
• Selmer group

• Lang, Serge (1983). Abelian Varieties. Springer Verlag. ISBN 3-540-90875-7.
• Mumford, David (1974). Abelian Varieties. Oxford University Press. ISBN 0-19-560528-4.