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 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 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.

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} }

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 .

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 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 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.