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nt.quantity principle – Constructing motivic representations via extensions of $mathrm{SL}(2, mathbb{Z})$ Answer

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nt.quantity principle – Constructing motivic representations via extensions of $mathrm{SL}(2, mathbb{Z})$

$ DeclareMathOperator SL {SL} DeclareMathOperator GL {GL} $By doing Sketch, Grothendieck explains how you can employ the representations of. finds $ G _ { mathbb {Q}} $ on Tate modules from Jacobians via his motion $ widehat { SL (2, mathbb {Z})} $. This has already been mentioned on this query: abelian $ ell $ -adic representations in $ widehat {SL (2, Z)} $.

In a footnote to this passage, Grothendieck mentions one route of increasing this thought $ ($Here, $ Gamma = G _ { mathbb {Q}}) $:

In 1981 I started experimenting with this machine in just a few particular circumstances and acquired varied preeminent representations of $ Gamma $ in teams $ G ( widehat { mathbb {Z}}) $, the place $ G $ is a (not essentially reductive) group strategy about $ mathbb {Z} $, ranging from apt homomorphisms $$ SL (2, mathbb {Z}) rightarrow G_0 ( mathbb {Z}), $$ Where $ G_0 $ a gaggle strategy is over $ mathbb {Z} $, and $ G $ is constructed as an extension of $ G_0 $ via a apt group strategy. In the “tautological” illustration $ G_0 = SL (2) _ { mathbb {Z}} $, we discover for $ G $ a grand extension of $ GL (2) _ { mathbb {Z}} $ by a torus of dimension 2, with a motivic illustration that “covers” these assigned to the category fields of the extensions $ mathbb {Q} (i) $ and $ mathbb {Q} (j) $ (How random the “fields of complex multiplication” of the 2 “anharmonic” elliptic curves).

I’m excited by judgement what this implies within the “tautological” illustration as properly. I can formulate the query as follows.

In common, how do you assemble a illustration (not essentially linear on this context, e.g. an exterior motion) of a apt extension? That is, with an require sequence of group schemes
$$ 1 rightarrow N rightarrow G rightarrow SL (2, mathbb {Z}) rightarrow 1, $$
one can take an motion from. bear $ G _ { mathbb {Q}} $ At $ G ( widehat { mathbb {Z}}) $ beginning with an motion by $ G _ { mathbb {Q}} $ At $ widehat { SL (2, mathbb {Z})} $? Break off:

  1. Even within the trifling illustration of $ N = 1 $, we’ve got this objective $ SL (2, mathbb {Z}) ( widehat { mathbb {Z}}) $ whose relationship to $ widehat { SL (2, mathbb {Z})} $ is unclear to me. Is it limpid that the presentation right here is basically what we began with?

  2. Assuming the primary drawback is solved, that $ N $ will of passage surrender an motion $ G ( widehat { mathbb {Z}}) $? Perhaps those that take an motion by themselves $ G _ { mathbb {Q}} $? In any illustration, Grothendieck appears to imply together with Tori.

  3. What does it denote for a motivic illustration to “cover” these which might be linked to the category fields of the extensions? $ mathbb {Q} (i) $ and $ mathbb {Q} (j) $? It sounds love there’s a several conception of motivic representations of sophistication fields (??) that I’m not close with, they usually can breathe realized as quotients of this newly constructed illustration.

  4. Why does Grothendieck say “a remarkable extension of $ GL (2) _ { mathbb {Z}} $“as an alternative of” a remarkable extension of $ SL (2) _ { mathbb {Z}} $“?

Any perception into these questions or explanations of higher inquiries to ameliorate grasp this passage would breathe vastly appreciated.

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