In my recent number theory seminar on “Hilbert’s 90 and generalizations” (notes here), Professor Goins asked the following interesting question.

Let {K} be a field and {d\in K^*}. Define {T_d} to be the torus

\displaystyle \left\{ \begin{pmatrix} x & dy \\ y & x \end{pmatrix} : x,y \in L, x^2-dy^2=1\right\}.

What values of {d} give {K}-isomorphic tori?

(The question was perhaps motivated by the observation that over the reals, the sign of {d} determines completely whether {T_d} would be split (i.e., isomorphic to {\mathbb R^*}) or anisotropic (i.e., isomorphic to {S^1}).

Here are two ways of looking at the answer.

  • For {d,e \in K^*}, we determine when two matrices {\displaystyle\begin{pmatrix} x & dy \\ y & x \end{pmatrix}} and {\displaystyle\begin{pmatrix} u & ev \\ v & u \end{pmatrix}} are conjugate in {\text{SL}_2(K)}. Solving the system

\displaystyle \begin{pmatrix} x & dy \\ y & x \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} . \begin{pmatrix} u & ev \\ v & u \end{pmatrix} . \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

gives {\displaystyle \frac{e}{d} = \left(\frac{b}{c}\right)^2, de = \left(\frac{d}{a}\right)^2}.

Thus {e \in d.(K^*)^2} i.e., the {T_d}‘s are classified by {\displaystyle \frac{K^*}{(K^*)^2}}. (For {K=\mathbb R}, this is isomorphic to {\{\pm 1\}} so the sign of {d} determines {T_d} upto conjugation.) By Kummer theory, {\displaystyle \frac{K^*}{(K^*)^2} \cong H^1(\text{Gal}(\overline K / K), \mu_2)}, where {\mu_2 = \{\pm 1\}} are the second roots of unity. Thus there is a correspondence between isomorphism classes of tori {T_d \; (d \in K^*)} and quadratic extensions of {K}.

  • Another way to look at the same thing is as follows. Fix {d \in K^*}. Let {L} be an extension of {K} wherein {T_d} splits. Now {T_d(L)} is a split torus of rank 1. For an algebraic group {G} over an algebraically closed field, we have the exact sequence

\displaystyle 1 \rightarrow \text{Inn}(G) \rightarrow \text{Aut}(G) \rightarrow \text{Aut}(\Psi_0(G)) \rightarrow 1,

where {\Psi_0(G)} is the based root datum {(X,\Delta,X\;\check{}, \Delta\,\check{})} associated to {G}. (Here, {X = X^*(G)} and {\Delta} is the set of simple roots of {X} corresponding to a choice of a Borel subgroup of {G}.) For details, see Corollary 2.14 of Springer’s paper “Reductive Groups” in Corvallis.

In our case, {G = T_d} so {\Psi_0(G) = ( \mathbb Z, \emptyset, \mathbb Z \;\check{}, \emptyset)}.

\displaystyle \text{Aut}(\Psi_0(G)) \cong \text{Aut}(\mathbb Z) \cong \{ \pm 1\}.

Now {L/K} forms of {T_d} are in bijective correspondence with

\displaystyle H^1(\text{Gal}(L/K), \text{Aut}(\Psi_0(G))) = H^1(\text{Gal}(L/K), \{\pm 1\}) \cong \text{Hom}_{\mathbb Z}(\text{Gal}(L/K), \{\pm 1\});

the last isomorphism because the Galois group acts trivially on the split torus {T_d(L)}. {\blacksquare}