# Character group of non-split torus in $GL_2$

Let $$E=\mathbb Q(\sqrt{-d})$$ be an imaginary quadratic field and let $$R_{E/\mathbb Q}(\mathbb G_m)$$ be the restriction of scalars of the multiplicative group, i.e. $$R_{E/\mathbb Q}(\mathbb G_m)(X) = \mathbb G_m(X \times_{\mathbb Q} E)$$ for each $$\mathbb Q$$-scheme $$X$$.

Picking a basis $$\langle 1, -\sqrt{-d}\rangle$$ for $$E$$ and letting $$E$$ act on itself, we can embed $$E$$ into $$M_2(\mathbb Q)$$ by $$(\alpha, \beta) \mapsto \begin{bmatrix}\alpha&-d\beta\\\beta&\alpha\end{bmatrix}$$.

Question 1: This embedding should give rise to an embedding of algebraic groups over $$\mathbb Q$$, $$R_{E/\mathbb Q}(\mathbb G_m) \hookrightarrow GL_2$$. Is $$R_{E/\mathbb Q}(\mathbb G_m)$$ a maximal (non-split) torus in $$GL_2$$? I seem to remember that the elements of maximal non-split tori in $$GL_n$$ all satisfy $$\det = 1$$, which is not the case here. What went wrong?

Question 2: Whatever the correct definition of the maximal torus in $$GL_2$$ obtained from $$E$$ is, how can we describe its character group explicitly?

Question 1: Yes, this is correct. In fact, in general, all maximal tori of $$\mathrm{GL}_{n,F}$$ are of the form $$\mathrm{Res}_{E/F}\mathbb{G}_{m,E}$$ where $$E$$ is an etale algebra over $$F$$ of degree $$n$$ (i.e. $$E=L_1\times\cdots\times L_n$$ with each $$L_i/F$$ a finite separable extension, and $$\sum_i [L_i:F]=n$$) (although you need to choose an embedding of $$E$$ into $$\mathrm{Mat}_{n,F}$$ to get your actual torus).

Question 2: In general if $$E=L_1\times\cdots\times L_n$$ then

$$X^\ast(\mathrm{Res}_{E/F}\mathbb{G}_{m,E})=\prod_i X^\ast(\mathrm{Res}_{L_i/F}\mathbb{G}_{m,L_i})$$

and $$X^\ast(\mathrm{Res}_{L_i/F}\mathbb{G}_{m,L_i})$$ is the permutation module for $$\mathrm{Gal}(\overline{F}/F)$$ associated to the $$\mathrm{Gal}(\overline{F}/F)$$-set $$\mathrm{Hom}_F(L_i,\overline{F})$$.