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The question is about whether one can view the p-adic local monodromy theorem as the quasi-unipotence of some monodromy operator.

it is known that the classical local monodromy theorem (i.e. for variation of Hodge structures) can be stated in terms of "linear algebra": let $\mathbb{V}$ be a polarized variation of Hodge structures on the punctured disc $\Delta^\times$. Then the monodromy operator $T$ is quasi-unipotent. Here the monodromy means the representation of the topological fundamental group on any fiber $V$ of $\mathbb{V}$, i.e. $\rho:\pi_1(\Delta^\times)\rightarrow GL(V)$, which is determined by the underlying local system $\mathbb{V}$. And this representation is simply characterized by $T=\rho(1)$ as one may identify $\pi_1(\Delta^\times)$ with $\mathbb{Z}$.

In the p-adic setting one has the Crew conjecture, also known as the p-adic local monodromy theorem, proved by Andre, Mebkhout, and Kedlaya. It can be stated as: let $F$ be the fraction field of a Witt ring of perfect residue of char.p, $R$ the Robba ring over $F$ (a ring of Lauent series convergent on some "thin" open annalus of outer radius 1 defined over $F$), and $M$ a differential module over $R$ with a compatible Frobenius structure (a slope zero $\phi$-module for the Frobenius structure on $R$ coming from the standard one on the Witt ring), then $M$ is quasi-unipotent, in the sense that after a finite extension of $R$ to some other Robba ring (coming from a finite extension of $F$), $M$ becomes unipotent, i.e. a successive extension of the trivial differential module $(R,d)$.

My question is whether one can find out a similar interpretation of quasi-unipotence. Namely, let $M$ be such a differential module with Frobenius structure on $R$, then show that it gives rise to a quasi-unipotent representation of the fundamental group of the annalus on a suitable space, possibly the solution space of $M$?

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  • $\begingroup$ it is understood that such differential module with compatible Frobenius structure is equivalent to a p-adic Galois representation, as is the starting point of the theory of $\phi,\Gamma$-modules of Fontaine, Colmez, etc. In that way does the unipotence of the differential modules implies any unipotence on the p-adic Galois representation? $\endgroup$
    – turtle
    Aug 25 '11 at 0:07
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    $\begingroup$ Colmez gave a Bourbaki talk on Les conjectures de monodromie $p$-adiques (Sém. Bourbaki 2001-02, exp. 897, Astérisque 290 (2003), 53--101, math.jussieu.fr/~colmez/897-asterisque.pdf). $\endgroup$ Aug 25 '11 at 3:54

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