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The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = 1$.

Question: why is this second generator called a monodromy, or, more precisely what O.D.E. corresponds to this situation?

Related question: Katz, in "On the calculation of some differential Galois groups" suggests to think of the inertia group as being analogous to the Tannaka group of some category of certain $D$-modules. How does that idea relate to the above situation?

Thank you!

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Dear Jakob -- In the analogous situation where the local field contains its residue field, such as $\mathbf{C}((t))$ or $\mathbf{F}_p((t))$, linear differential equations in the uniformizer are closely related to Galois theory, or at least the tame part. $\tau$ plays the role of a loop around a punctured disk. – JBorger Jun 15 '13 at 3:04

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