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I think this should be indeed related to motives; quite a lot is known about the motives of abelian varieties. I'll look up the(update: I think others provided some good references.)
ilya's comment: I think it would be very useful if somebody posted a question along the lines of what Thomas suggests, especially filling in some background from Illusie's paper (I would do it, but I don't have the paper itself).
I think this should be indeed related to motives; quite a lot is known about the motives of abelian varieties. I'll look up the references.
I think this should be indeed related to motives. (update: I think others provided some good references.)
ilya's comment: I think it would be very useful if somebody posted a question along the lines of what Thomas suggests, especially filling in some background from Illusie's paper (I would do it, but I don't have the paper itself).
The great references given above let me ask about the current status of the many conjectures and open questions in Illusie's survey, e.g. finiteness theorems, crystalline coefficients, geometric semistability,... ?
a further question by Thomas:
The great references given above let me ask about the current status of the many conjectures and open questions in Illusie's survey, e.g. finiteness theorems, crystalline coefficients, geometric semistability,... ?
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