# Isogeny

An epimorphism of group schemes (cf. Group scheme) with a finite kernel. A morphism of group schemes over a ground scheme is said to be an isogeny if is surjective and if its kernel is a flat finite group -scheme.

In what follows it is assumed that is the spectrum of a field of characteristic . Suppose that is a group scheme of finite type over , and let be a finite subgroup scheme. Then the quotient exists, and the natural mapping is an isogeny. Conversely, if is an isogeny of group schemes of finite type and , then . For every isogeny of Abelian varieties there exists an isogeny such that the composite is the homomorphism of multiplication of by . Composites of isogenies are isogenies. Two group schemes and are said to be isogenous if there exists an isogeny . An isogeny is said to be separable if is an étale group scheme over . This is equivalent to the fact that is a finite étale covering. An example of a separable isogeny is the homomorphism , where . If is a finite field, then every separable isogeny of connected commutative group schemes of dimension one factors through the isogeny , where and is the Frobenius endomorphism. An example of a non-separable isogeny is the homomorphism of multiplication by in an Abelian variety .

Localization of the additive category of Abelian varieties over with respect to isogeny determines an Abelian category , whose objects are called Abelian varieties up to isogeny. Every such object can be identified with an Abelian variety , and the morphisms in are elements of the algebra over the field of rational numbers. An isogeny defines an isomorphism of the corresponding objects in . The category is semi-simple: each of its objects is isomorphic to a product of indecomposable objects. There is a complete description of when is a finite field (see [4]).

The concept of an isogeny is also defined for formal groups. A morphism of formal groups over a field is said to be an isogeny if its image in the quotient category of the category of formal groups over by the subcategory of Artinian formal groups is an isomorphism. An isogeny of group schemes determines an isogeny of the corresponding formal completions. There is a description of the category of formal groups up to isogeny (see [1], [2]).

#### References

 [1] Yu.I. Manin, "The theory of commutative formal groups over fields of finite characteristic" Russian Math. Surveys , 18 : 6 (1963) pp. 1–81 Uspekhi Mat. Nauk , 18 : 6 (1963) pp. 3–90 [2] D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974) [3] J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) [4] J.T. Tate, "Classes d'isogénie des variétés abéliennes sur un corps fini (d' après T. Honda)" , Sem. Bourbaki Exp. 352 , Lect. notes in math. , 179 , Springer (1968/69) [5] J. Dieudonné, "Groupes de Lie et hyperalgèbres de Lie sur un corps de characteristique " Comm. Math. Helvetici , 28 : 1 (1954) pp. 87–118