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 characteristique " Comm. Math. Helvetici , 28 : 1 (1954) pp. 87–118