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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 $E/F$ ?

Paul Broussous
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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 09 '10 at 14:39
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    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 09 '10 at 14:53
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    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 09 '10 at 16:26
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    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 09 '10 at 16:35
  • @B: (Peter who??) Of course lots of people are confused by the fact that an elliptic curve is not an ellipse, but not people who are actually studying elliptic curves. I imagine the same would hold here. (To be fair, the phrase "complex torus" is even worse.) – Pete L. Clark Dec 09 '10 at 16:49
  • @Jim Humphreys. The problem arises indeed in the context of local fields. – Paul Broussous Dec 09 '10 at 18:06
  • @Pete L. Clark. The term "elliptic torus" is standard in the theory of automorphic forms. – Paul Broussous Dec 09 '10 at 18:09
  • @BCnrd OK, but I just miss the following argument. If $T$ is an anisotropic subtorus of ${\rm{Res}}{E/F}(\mathbf{G}_m)$, why do we have $N{E/F}; (T)=1$ ? – Paul Broussous Dec 09 '10 at 18:16
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    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 09 '10 at 18:53
  • @BCnrd It turns out that I need the non-maximal ones as well. – Paul Broussous Dec 15 '10 at 08:19

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