Tori over Finite Fields

Take \(k = \mathbb{F}_q\). The absolute Galois group is procyclic, \(\Theta = \widehat{\mathbb{Z}}\), topologically generated by the Frobenius \(\varphi : x \mapsto x^q\). By F-Tori Are Classified by Galois Modules (in the cocycle form Tori Are Classified by Cocycles in GL_n(Z)):

Rank-\(n\) tori over \(\mathbb{F}_q\) are in bijection with conjugacy classes of elements of finite order in \(\operatorname{GL}_n(\mathbb{Z})\).

Indeed a continuous homomorphism \(\widehat{\mathbb{Z}} \to \operatorname{GL}_n(\mathbb{Z})\) is determined by the image \(F\) of \(\varphi\); continuity (open kernel, The Minimal Splitting Field of a Torus) forces \(F\) to have finite order, and conversely any finite-order \(F\) determines a continuous homomorphism. Conjugacy of homomorphisms is conjugacy of \(F\).

The torus with Frobenius datum \(F\) of order \(d\) splits over \(\mathbb{F}_{q^d}\), and its group of rational points has order \[ \bigl|T(\mathbb{F}_q)\bigr| \;=\; \bigl|\det\nolimits_{X_*}(q\,F - 1)\bigr|, \] e.g. the norm-one torus of \(\mathbb{F}_{q^2}/\mathbb{F}_q\) (datum \(F = -1\) in \(\operatorname{GL}_1\)) has \(q + 1\) points.

Remarks.

• Every torus over a finite field is quasi-split in the trivial sense (it is a torus), but more is true: by Lang's theorem (Lang Theorem for Connected Algebraic Groups over Finite Fields) every torsor under a connected group is trivial, which is why the classification here has no second cohomological layer.
• This bijection underlies the parametrization of the maximal tori of a finite reductive group \(G(\mathbb{F}_q)\) by \(F\)-conjugacy classes in its Weyl group (Deligne–Lusztig theory); the case treated here is \(G = \mathbb{G}_m^{\,n}\).