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removed a false statement
Bhargav
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For the last question: at least in the algebro-geometric case, the monodromy is always quasi-unipotent (i.e, some power of is unipotent). There is a beautiful argument due to Grothendieck that proves this by reducing to the p-adic case, and using the action of Frobenius on the (tame) inertia group of a p-adic field.

Bhargav
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