Isogeny

An epimorphism of group schemes (cf. Group scheme) with a finite kernel. A morphism $ f: G \rightarrow G _ {1} $ of group schemes over a ground scheme $ S $ is said to be an isogeny if $ f $ is surjective and if its kernel $ \mathop{\rm Ker} ( f ) $ is a flat finite group $ S $- scheme.

In what follows it is assumed that $ S $ is the spectrum of a field $ k $ of characteristic $ p \geq 0 $. Suppose that $ G $ is a group scheme of finite type over $ k $, and let $ H $ be a finite subgroup scheme. Then the quotient $ G/H $ exists, and the natural mapping $ G \rightarrow G/H $ is an isogeny. Conversely, if $ f: G \rightarrow G _ {1} $ is an isogeny of group schemes of finite type and $ H = \mathop{\rm ker} ( f ) $, then $ G _ {1} = G/H $. For every isogeny $ f: G \rightarrow G _ {1} $ of Abelian varieties there exists an isogeny $ g: G _ {1} \rightarrow G $ such that the composite $ g \circ f $ is the homomorphism $ n _ {G} $ of multiplication of $ G $ by $ n $. Composites of isogenies are isogenies. Two group schemes $ G $ and $ G _ {1} $ are said to be isogenous if there exists an isogeny $ f: G \rightarrow G _ {1} $. An isogeny $ f: G \rightarrow G _ {1} $ is said to be separable if $ \mathop{\rm ker} ( f ) $ is an étale group scheme over $ k $. This is equivalent to the fact that $ f $ is a finite étale covering. An example of a separable isogeny is the homomorphism $ n _ {G} $, where $ ( n, p) = 1 $. If $ k $ is a finite field, then every separable isogeny $ f: G \rightarrow G _ {1} $ of connected commutative group schemes of dimension one factors through the isogeny $ \mathfrak p: G \rightarrow G $, where $ \mathfrak p = F - \mathop{\rm id} _ {G} $ and $ F $ is the Frobenius endomorphism. An example of a non-separable isogeny is the homomorphism of multiplication by $ n = p ^ {r} $ in an Abelian variety $ A $.

Localization of the additive category $ A ( k) $ of Abelian varieties over $ k $ with respect to isogeny determines an Abelian category $ M ( k) $, whose objects are called Abelian varieties up to isogeny. Every such object can be identified with an Abelian variety $ A $, and the morphisms $ A \rightarrow A _ {1} $ in $ M ( k) $ are elements of the algebra $ \mathop{\rm Hom} _ {A ( k) } ( A, A _ {1} ) \otimes _ {\mathbf Z } \mathbf Q $ over the field of rational numbers. An isogeny $ f: A \rightarrow A _ {1} $ defines an isomorphism of the corresponding objects in $ M ( k) $. The category $ M ( k) $ is semi-simple: each of its objects is isomorphic to a product of indecomposable objects. There is a complete description of $ M ( k) $ when $ k $ is a finite field (see [4]).

The concept of an isogeny is also defined for formal groups. A morphism $ f: G \rightarrow G _ {1} $ of formal groups over a field $ k $ is said to be an isogeny if its image in the quotient category $ \phi ( k) $ of the category of formal groups over $ k $ 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 $ \phi ( k) $ 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 caractéristique $p > 0$" Comm. Math. Helvetici , 28 : 1 (1954) pp. 87–118
[a1]	T. Honda, "Isogeny classes of Abelian varieties over finite fields" Math. Soc. Japan , 20 (1968) pp. 83–95
[a2]	J. Tate, "Endomorphisms of Abelian varieties over finite fields" Invent. Math. , 2 (1966) pp. 134–144