Galois descent, twisted forms, and inner forms

Let $K/k$ be a finite Galois extension with group $\Gamma$ , and let $H/K$ be an algebraic group. A $K/k$ -descent datum is a family of semilinear isomorphisms

\phi_\sigma:{}^\sigma H\longrightarrow H

satisfying $\phi_{\sigma\tau}=\phi_\sigma\circ{}^\sigma\phi_\tau$ . For affine algebraic groups, finite Galois descent is effective.

Forms

Fix a $k$ -group $G_0$ . Forms of $G_0$ split by $K$ , after choosing a $K$ -identification with $(G_0)_K$ , are classified by

H^1\!\left(\Gamma,\operatorname{Aut}(G_0)(K)\right).

Changing the identification changes the cocycle by nonabelian cohomology.

Inner and pure inner twists

An inner form has class in the image of

H^1(k,G_{0,\mathrm{ad}}) \longrightarrow H^1(k,\operatorname{Aut}(G_0)).

A pure inner twist consists of an inner twisting isomorphism $\psi:G_0\to G'$ over $k_s$ together with a cocycle $z\in Z^1(k,G_0)$ satisfying

\psi^{-1}\sigma(\psi)=\operatorname{Int}(z_\sigma).

Thus a cocycle in $G_0$ is part of the data; it is not merely a name for the resulting $k$ -group. Not every inner form admits a pure inner twist.

The refined local Langlands correspondence often needs the still more flexible rigid rigid inner-twist formalism.

Relation to the letter

The letter first descends a split group through outer automorphisms and then modifies it by an inner class. Its requirement that the inner class become locally trivial at almost all places is the precursor of the global organization of local inner forms.

References

Jean-Pierre Serre, Galois Cohomology, Springer, 1997.
Tasho Kaletha, “Rigid inner forms of real and $p$ -adic groups,” Annals of Mathematics 184 (2016), 559–632. DOI.