# anisotropic and elliptic tori in GL(n)

Let $F$ be a commutative field and $n\geqslant 2$ be an integer. It is well known that the maximal anisotropic mod center tori in $G={\rm GL}(n,F)$ are of the form $T = {\rm Res}_{E/F}\; {\mathbb G}_m$,for some degree $n$ separable field extension $E/F$, where ${\rm Res}$ denotes Weil's restriction of scalar and where ${\mathbb G}_m$ denotes the $1$-dimensional split torus.

Such a torus $T$ embeds in $G$ in the following way. One identifies $G$ with ${\rm Aut}_F\; (E)$ and make $E^{\times} =T(F)$ acts on $E$ by multiplication.

One says that a torus is elliptic if it is not contained in any proper parabolic subgroup of $G$. The tori $T$ described above are elliptic.

My question is :

Is it true that the subtori of $T$ that are elliptic and anisotropic are of the form : $S=\{ t\in T\ ; \ N_{E/L}(t)=1\}$, where $L/F$ is a proper subfield extension of $E/F$ ?

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It would help me to have a reference or two to basic sources of this terminology. My impression is that the notion of "elliptic torus" has only been studied over local fields, whereas "anisotropic torus" occurs more widely. What is the context of your question? – Jim Humphreys Dec 9 '10 at 14:39
The max'l $F$-tori are $T_E := {\rm{Res}}_{E/F}(\mathbf{G}_m)$ for finite etale $F$-algebras $E$ of degree $n$, embedded via an ordered $F$-basis of $E$. Note $E = \prod E_i$ with fields $E_i$, so $T_E = \prod T_{E_i}$ and the max'l $F$-anisotropic subtorus is the product $\prod T_{E_i}^{1}$ of norm-1 subtori of factors. If at least 2 factor fields, it lies in a proper parabolic $F$-subgp, so not elliptic. Thus, elliptic anisotropic tori are contained in the elliptic anisotropic $T_E^1$ with $E/F$ a degree-$n$ sepble field extn. Do you really want the non-maximal examples too? – BCnrd Dec 9 '10 at 14:53
Am I the only one who thinks that using the term "elliptic torus" for a linear algebraic group is essentially begging to be misunderstood? – Pete L. Clark Dec 9 '10 at 16:26
Dear Peter: The terminology is entirely standard, and despite knowing about tori and elliptic curves for a long time, it never crossed my mind that the phrase "elliptic torus" could create a misunderstanding. It is no worse than the fact that an elliptic curve is not an ellipse. So you may be the only one. :) – BCnrd Dec 9 '10 at 16:35
Dear Paul: an anisotropic torus over a field has no nontrivial homomorphisms to $\mathbf{G}_m$ over the field, whence an anisotropic subtorus must be killed by the norm character. – BCnrd Dec 9 '10 at 18:53