In mathematics, particularly in algebraic geometry, an isogeny between two abelian varieties is a surjective homormophism with a finite kernel. In the case of abelian varieties, then any morphism f : A → B of the underlying algebraic varieties which is surjective with finite fibres is automatically an isogeny, provided that f(1A) = 1B. Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over which 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; before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny.
Degree of 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 the mapping is nontrivial, it is a field embedding and im f ∗ {\displaystyle \operatorname {im} f^{*}} is a subfield of K(A). The degree 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:
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}
If c h a r K ∤ deg f {\displaystyle char\;K\nmid \deg f} , then deg f = | ker f | {\displaystyle \deg f=|\ker \;f|}
Case of abelian varieties
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.
Tate's isogeny theorem In mathematics, Tate's isogeny theorem, proved by Tate (1966), states that two abelian varieties over a finite field are isogeneous if and only if their Tate modules are isomorphic (as Galois representations).
See also Abelian varieties up to isogeny Selmer group Honda–Tate theorem
Notes
References Milne, James (2008). Abelian Varieties (PDF). Retrieved 2026-07-27. Weil, André (1948). Variétés abéliennes et courbes algébriques. Hermann. Lang, Serge (1983). Abelian Varieties. Springer Verlag. ISBN 3-540-90875-7. Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 9788185931869, MR 0282985, OCLC 138290 Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae, 2 (2): 134–144, Bibcode:1966InMat...2..134T, doi:10.1007/BF01404549, ISSN 0020-9910, MR 0206004, S2CID 245902
External links Mumford, David (1974). Abelian Varieties. Oxford University Press. ISBN 0-19-560528-4.
