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?


1 Answer 1


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 $\mathrm{Hom}_F(L_i,\overline{F})$.


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