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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 module 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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  • $\begingroup$ 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. $\endgroup$
    – Daniel Larsson
    Commented Feb 17, 2012 at 17:23
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    $\begingroup$ 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.) $\endgroup$
    – Daniel Larsson
    Commented Feb 17, 2012 at 17:28

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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 $\mathbb{Z}_p\cap\mathbb{Q}$.

  • 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 published in the "rendiconti" of Padua.

For both of these notions you can consult the Book (under construction) of G.Christol Le théorème de Turrittin $p$-adique.

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 Filtrations de type Hasse–Arf et monodromie $p$-adique (section 7.2). This conjecture has been proved by Z.Mebkhout, you may find at the bottom of this post the reference.

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 completely classified these equations, this provides an elementary but quite intricate class of examples) Rank one solvable $p$-adic differential equations and finite Abelian characters via Lubin–Tate groups.

@InCollection{zbMATH05504068, Author = {Mebkhout, Zoghman}, Title = {Frobenius structures and monodromy exponents of {{(p)}}-adic differential equations}, BookTitle = {Singularities I. Algebraic and analytic aspects. Proceedings of the international conference ``School and workshop on the geometry and topology of singularities'' in honor of the 60th birthday of L^e D~ung Tr'ang, Cuernavaca, Mexico, January 8--26, 2007}, ISBN = {978-0-8218-4458-8}, Pages = {175--243}, Year = {2008}, Publisher = {Providence, RI: American Mathematical Society (AMS)}, Language = {French}, Keywords = {12H25,14F40,11S20}, zbMATH = {5504068}, Zbl = {1175.12002} }

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  • $\begingroup$ I'm working on effectively computing generic radii in this context (building on your CRAS note An algorithm computing non-solvable spectral radii of $p$-adic differential equations). I believe that the proof of this conjecture by Z. Mebkhout could eventually provide an effective test to know if the algorithm proposed in your note has ended, assuming the rationality of exponents. Could you confirm if it has indeed been proved since 2012 ? I can't find any reference for such a proof. $\endgroup$
    – Lucas
    Commented Mar 18 at 22:48
  • $\begingroup$ The links had rotted, so I tried filtering them through the Wayback Machine. The one to Christol's page re-directed to a site that itself seems only accessible by Wayback Machine; I left it as a Wayback link to a re-direct, although I think that Christol - Le théorème de Turrittin $p$-adique is probably the correct link. I didn't find the paper by Dwork you referenced, but I wonder if its contents are identical to, or overlap with, Dwork - On exponents of $p$-adic differential modules. $\endgroup$
    – LSpice
    Commented Mar 18 at 23:46
  • $\begingroup$ This is the reference where Mebkhout proved Dwork conjecture : Mebkhout, Zoghman}, Title = {Frobenius structures and monodromy exponents of {{(p)}}-adic differential equations}, BookTitle = {Singularities I. Algebraic and analytic aspects. Proceedings of the international conference ``School and workshop on the geometry and topology of singularities'' in honor of the 60th birthday of L\^e D\~ung Tr\'ang, Cuernavaca, Mexico, January 8--26, 2007}, ISBN = {978-0-8218-4458-8}, Pages = {175--243}, Year = {2008}, Publisher = {Providence, RI: American Mathematical Society (AMS)}, $\endgroup$
    – PULITA ANDREA
    Commented Mar 20 at 0:49
  • $\begingroup$ In the following article he gives an idea of the steps of the proof Author = {Mebkhout, Zoghman}, Title = {On the monodromy theorem for the family of {{(p)}}-adic differential equations}, FJournal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France}, Journal = {Bull. Soc. Math. Fr.}, ISSN = {0037-9484}, Volume = {148}, Number = {4}, Pages = {651--708}, Year = {2020}, DOI = {10.24033/bsmf.2820}, $\endgroup$
    – PULITA ANDREA
    Commented Mar 20 at 0:51
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    $\begingroup$ Yes this is the right link $\endgroup$
    – PULITA ANDREA
    Commented Mar 20 at 0:51

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