ISOGENY

Etymology

Noun

isogeny (countable and uncountable, plural isogenies)

The condition of being isogenous.

(algebraic geometry, category theory) An epimorphism of group schemes that is surjective and has a finite kernel.

$\backslash (\backslash varphi\backslash circ\backslash hat\{\backslash varphi\}=\backslash left\; [\backslash operatorname\{deg\}(\backslash varphi)\backslash right\; ]=\backslash left\; [\backslash operatorname\{deg\}(\backslash hat\{\backslash varphi\})\backslash right\; ]\backslash )$,

so that $\backslash (\backslash varphi\backslash )$ is in turn the dual isogeny of its dual $\backslash (\backslash hat\{\backslash varphi\}\backslash )$. Isogeny of complex tori, rather than isomorphism, will turn out to be the appropriate equivalence relation in the context of modular forms.

Usage notes

In some contexts, (e.g, universal algebra), an epimorphism may be defined as a surjective homomorphism, and the definition of isogeny may change accordingly. In the broader context of category theory, however, this substitution is not made, because the definitions are not precisely identical. (A surjective homomorphism is always an epimorphism, but the reverse is not always true. See )

Source: Wiktionary

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