Rational Forms of Linear Algebraic Groups

A rational form of a linear algebraic group is a \(k\)-form: a linear algebraic group over \(k\) becoming isomorphic to a fixed geometric model after base change. The classification by Galois cohomology applies to arbitrary linear algebraic groups, not just the reductive case.

Definition

Let \(G\) be a linear algebraic group over a separably closed field \(\bar k\). A rational form of \(G\) over \(k\) is a linear algebraic \(k\)-group \(G'\) such that \(G' \otimes_k \bar k \cong G\).

Classification

\(G'\)-isomorphism classes of forms of \(G\) over \(k\) are in bijection with \[ H^1(k, \mathrm{Aut}(G)). \]

This works for any linear algebraic group \(G\): reductive, unipotent, solvable, etc.

Examples

Tori (specialisation to abelian groups)

A torus \(T\) over \(k\) becomes split over a finite Galois extension \(K/k\) with Galois group \(\Gamma'\). Forms of the split torus \(\mathbb{G}_m^n\) are classified by: \[ H^1(\Gamma', \mathrm{Aut}(\mathbb{G}_m^n)) \;=\; H^1(\Gamma', \mathrm{GL}_n(\mathbb{Z})), \] i.e.\ continuous representations \(\Gamma' \to \mathrm{GL}_n(\mathbb{Z})\) up to conjugation. This recovers the torus-lattice equivalence.

Unipotent groups

\(\mathbb{G}_a\) has no non-trivial forms over a perfect field \(k\) (\(\mathrm{Aut}(\mathbb{G}_a) = \mathbb{G}_m\) acts by scaling, and \(H^1(k, \mathbb{G}_m) = 0\) by Hilbert 90 — so the only form is \(\mathbb{G}_a\) itself).

Over non-perfect fields of characteristic \(p\): there can be "twisted" forms of \(\mathbb{G}_a\) that are not isomorphic to \(\mathbb{G}_a\) over \(k\) — but such forms are not smooth.

Solvable groups

For the "\(ax + b\)" group \(\mathbb{G}_a \rtimes \mathbb{G}_m\): a form is a non-split extension; classified by \(H^1(k, \mathrm{Aut})\) = \(\mathbb{G}_a \rtimes \mathbb{G}_m\) up to inner twisting.

Reductive groups

The largest interesting class: see reductive forms.

Properties

For \(G\) a smooth linear algebraic group over \(k\):

Forms of \(G\) are smooth and of the same dimension.
Forms have the same identity component as their base change.
The set of forms can be infinite (e.g.\ all central simple algebras for \(\mathrm{GL}_n\)).

For \(G\) commutative: forms classified by an honest abelian group \(H^1(k, G)\) (the group structure of forms is inherited).

For \(G\) non-commutative: forms classified by a pointed set \(H^1(k, G)\); the "trivial form" is the basepoint.

Local-global principle

For a global field \(F\), an important question is the Hasse principle:

\[ \Sha^1(F, G) \;:=\; \ker(H^1(F, G) \to \prod_v H^1(F_v, G)) \;\overset{?}{=}\; 0. \]

For \(G\) commutative: \(\Sha^1\) is often non-trivial (Tate- Shafarevich groups of tori, of abelian varieties...).
For \(G\) connected reductive simply connected: \(\Sha^1 = 0\) (Kneser-Harder-Chernousov; see Galois cohomology).
For non-simply-connected \(G\): \(\Sha^1\) is computable via Borovoi abelianisation.

Twisting and forms

A 1-cocycle \(c \in Z^1(k, \mathrm{Aut}(G))\) determines a twisted form \(G^c\). The construction is functorial:

An exact sequence \(1 \to H \to G \to Q \to 1\) gives \(1 \to H^c \to G^c \to Q^c \to 1\), with each group twisted by the appropriate cocycle.

This is the technical basis for transferring questions between different forms (e.g.\ via inner twists).