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Author: William A. Stein
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Dear Ken,
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> p. 10 (+7) Perhaps explain why this is a concrete consequence,
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> saying a little about J_0(Nl^2); also, the end of the line
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> sticks out badly into the margin.
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That this is a concrete consequence is explained in Section 3.1 of chapter 3.
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Why does Brian want level Nl^2 instead of level Nl?
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RIBET: If you want weights beyond l+1, then you have to twist
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by powers of the cyclotomic character, which you view as a character
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of conductor l. That twisting raises level from ...l to ...l^2.
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Whether Brian is right or not depends on the context -- were
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high weights contemplated at that juncture?
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WAS: -- YES -- It says:
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A concrete consequence of the conjecture is that all odd
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irreducible 2-dimensional~$\rho$ come from abelian varieties
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over~$\Q$. Given~$\rho$, one should be able to find a totally
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real or CM number field~$E$, an abelian variety~$A$ over~$\Q$
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of dimension [...]
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I now see why it is necessary to introduce ell^2. However,
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I'm not sure we should explain further, at least at the late point
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in the writing of the paper.
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*However* maybe we should mention "Theorem F" (page 4) of Taylor's amazing new
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paper "Remarks on a Conjecture of Fontaine and Mazur", which seems to prove the
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above concrete consequence under a reasonable local hypothesis.
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William
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