nLab isogeny (changes)

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Context

Group Theory

Idea

References

Idea

An isogeny is a surjection of algebraic groups that has a finite kernel. For abelian varieties, this is equivalently a rational map between groups of the same dimension that preserves the neutral element (which implies that it is a group homomorphism).

Lang isogeny

Let $G$ be a commutative algebraic group over a finite field $\mathbf{F}_q$ , and write $\mathrm{Fr}$ for its $q$ -power Frobenius. The Lang isogeny is the morphism

(1)

\mathrm{Lang} : G \to G \qquad g \mapsto \mathrm{Fr}(g) \cdot g^{-1}.

Observe that

(2)

\mathrm{ker}(\mathrm{Lang}) = G(\mathbf{F}_q).

One may view the Lang isogeny as defining a multiplicative $G(\mathbf{F}_q)$ -local system over $G$ . It follows that every character

(3)

\chi : G(\mathbf{F}_p) \to \overline{\mathbf{Q}}_\ell^\times

induces a character sheaf $\mathscr{F}_\chi$ on $G$ , and one verifies that applying the fonctions-faisceaux dictionary? to $\mathscr{F}_\chi$ recovers $\chi$ .

References

This Stack Exchange discussion, What is isogeny?
Wikipedia, Isogeny

Examples of sporadic (exceptional) isogenies from spin groups onto orthogonal groups are discussed in

Paul Garrett, Sporadic isogenies to orthogonal groups (July 2013) [[pdf](http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf), pdf ]

Last revised on August 1, 2024 at 01:41:18. See the history of this page for a list of all contributions to it.