isogeny

English

Etymology

iso- +‎ -geny.

Noun

isogeny (countable and uncountable, plural isogenies)

1. The condition of being isogenous.
2. (algebraic geometry, category theory) An epimorphism of group schemes that is surjective and has a finite kernel.
   • 2000, Marc Hindry, Joseph H. Silverman, Diophantine Geometry: An Introduction, Springer, page 95:

     It is clear that if $G_{2}$ is connected, then two of the defining properties of an isogeny imply the third.

   • 2002, Mireille Fouquet, François Morain, Isogeny Volcanoes and the SEA Algorithm, Claus Fieker, David R. Kohel (editors), Algorithmic Number Theory: 5th International Symposium, Proceedings, Springer, LNCS 2369, page 279,

     Lemma 2.2 Let $E$ be an elliptic curve such that $\mathbb {Z} [\pi ]$ is maximal at ${\mathcal {l}}$. If there exists an ${\mathcal {l}}$-isogeny of $E$, then this ${\mathcal {l}}$-isogeny is an^[sic] horizontal ${\mathcal {l}}$-isogeny.

   • 2005, Fred Diamond, Jerry Shurman, A First Course in Modular Forms, Springer, page 29,

     The dual isogeny of an isomorphism is its inverse. The dual of a composition of isogenies is the composition of the duals in the reverse order. If $\varphi$ is an isogeny and ${\hat {\varphi }}$ is its dual then the formulas $\varphi (z+\Lambda )=mz+\Lambda '$, $\varphi (z'+\Lambda ')=(\operatorname {\varphi } /m)z'+\Lambda$ show that also

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

     so that $\varphi$ is in turn the dual isogeny of its dual ${\hat {\varphi }}$. 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 Epimorphism on Wikipedia.Wikipedia )

Derived terms

• dual isogeny
• isogenic
• isogenous
• isogeny cycle
• isogeny graph
• isogeny volcano

Translations

epimorphism of group schemes that is surjective and has a finite kernel

• Finnish: isogenia

Further reading

• Isogeny on Encyclopedia of Mathematics
• Isogeny on Wolfram MathWorld