Timeline for anisotropic and elliptic tori in GL(n)
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Dec 15, 2010 at 8:19 | comment | added | Paul Broussous | @BCnrd It turns out that I need the non-maximal ones as well. | |
| Dec 15, 2010 at 8:18 | history | edited | Paul Broussous | CC BY-SA 2.5 |
I've made my question more precise
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| Dec 9, 2010 at 18:53 | comment | added | BCnrd | 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. | |
| Dec 9, 2010 at 18:16 | comment | added | Paul Broussous | @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 $N_{E/F}\; (T)=1$ ? | |
| Dec 9, 2010 at 18:09 | comment | added | Paul Broussous | @Pete L. Clark. The term "elliptic torus" is standard in the theory of automorphic forms. | |
| Dec 9, 2010 at 18:06 | comment | added | Paul Broussous | @Jim Humphreys. The problem arises indeed in the context of local fields. | |
| Dec 9, 2010 at 16:49 | comment | added | Pete L. Clark | @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.) | |
| Dec 9, 2010 at 16:35 | comment | added | BCnrd | 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. :) | |
| Dec 9, 2010 at 16:26 | comment | added | Pete L. Clark | Am I the only one who thinks that using the term "elliptic torus" for a linear algebraic group is essentially begging to be misunderstood? | |
| Dec 9, 2010 at 14:53 | comment | added | BCnrd | 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? | |
| Dec 9, 2010 at 14:39 | comment | added | Jim Humphreys | 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? | |
| Dec 9, 2010 at 11:59 | history | asked | Paul Broussous | CC BY-SA 2.5 |