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Let $\mathcal{E}$ be the ring of series $f(T)=\sum_{n \in \mathbb{Z}} a_nT^n, \; a_n \in \mathbb{Q}_p$ such that the $\{a_n\}$ are bounded and $a_{-n}\rightarrow 0$ as $n \rightarrow +\infty$. This is the usual coefficient ring in Fontaine's theory of $(\varphi, \Gamma)$-modules. This ring comes equipped with a Frobenius endomorphism $\varphi$ and a derivation $\frac{d}{dT}$. I recently read that any finite free module over $\mathcal{E}$ equipped with a unit-root $\varphi$-structure automatically admits a unique compatible differential modules structure. Why is this true?

Conversely, if we take a finite free module $M$ over $\mathcal{E}$ equipped with a differential module structure can we find a compatible unit-root Frobenius structure on $M$? Furthermore, if we can find one, to what extent is it unique?

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I think this is somewhere in Kiran Kedlaya's recent book on $p$-adic differential equations. I don't have it nearby so I don't know for sure. Otherwise, have a look at some of Laurent Berger's surveys. You might find something there. –  Daniel Larsson Feb 17 '12 at 17:23
    
Oh, another thing you might want to give a (loose) definition of $\mathcal{E}$, because there are so many of "Fontaine's rings" (and not all by Fontaine); what's worse different notations for the same rings. (I assume you mean the "Robba ring", but I could be wrong.) –  Daniel Larsson Feb 17 '12 at 17:28
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1 Answer

Hi,

The fact that the Frobenius produces a differential equation is classical. Dwork was the first to find it out. As an reference in your exact context you have the paper of Fontaine in the Grothendieck Festschrift Vol.2, section 2.2.4.

The converse of this fact is false, but there exists a conjecture of Dwork about it (see below).

1) If you have a connection you possibly do not have any Frobenius compatible to it in general (below I provide an example).

2) Moreover this Frobenius is never unique since a Frobenius is basically an isomorphism between the differential equation and its pull-back by the Frobenius functor. The family of morphisms between two differential equations form a vector space (the category is always additive, and depending on the type of equations that you consider it can be often abelian and Tannakian).

EXAMPLE : Consider the equation $y' = (\frac{a}{x})\cdot y$, where $a$ is a constant element, and $x$ is the variable. It is known that this equation has a Frobenius structure if and only if $a$ lies in $\mathbb{Z}_p\cap\mathbb{Q}$.

$\bullet$ The existence of a Frobenius structure implies some restriction on the equation. The major of them are the solvability and the rationality of the so called exponents.

SOLVABILITY : This is a condition about the radius of convergence of the solutions at the generic point (this corresponds to a certain point of a certain Berkovich space). Solvability means that at this particular point the taylor solutions of the equations have the largest possible radius. Robba proved that the equation of the above example is solvable if and only if $a\in\mathbb{Z}_p$.

EXPONENTS : The definition of the $p$-adic exponents is quite complicated. They have been defined firstly by G.Christol, then improved (i.e. simplified) by B.Dwork publiched in the "rendiconti" of Padua.

For both of these notions you can consult the Book (under construction) of G.Christol http://www.math.jussieu.fr/~christol/courspdf.pdf

THE CONJECTURE : B.Dwork conjectured that under these two conditions solvability + rationality of the exponents the equation should have a so called Frobenius structure (i.e. the existence of a Frobenius). This conjecture have been the object of a section in the paper of Y.André http://arxiv.org/abs/math/0203248 (section 7.2). This conjecture seems to have been (very recently) proved by Z.Mebkhout.

NOTE : The conjecture is true for rank one differential modules over the Robba ring. This is due to B.Chiarellotto G.Christol in the case of polynomial coefficients, and to myself in the general case (I also have copletely classified these equations, this provides an elementary but quite intricated class of examples) http://www.math.univ-montp2.fr/~pulita/Publications/Rk1.pdf .

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