nLab isogeny

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

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

Observe that

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

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

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

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, pdf ]