Infinitesimal deformation of p-adic differential equations on Berkovich curves

We show that if a differential equations $mathscr{F}$ over a quasi-smooth Berkovich curve $X$ has a certain compatibility condition with respect to an automorphism $sigma$ of $X$, and if the automorphism is sufficiently close to the identity, then $mathscr{F}$ acquires a semi-linear action of $sigma$ (i.e. lifting that on $X$). This generalizes the previous works of Yves Andre, Lucia Di Vizio, and the author about $p$-adic $q$-difference equations. We also obtain an application to Moritas $p$-adic Gamma function, and to related values of $p$-adic $L$-functions.

Solvability of rank one $p$-adic differential and $q$-difference equations over the Amice ring

We provide a necessary and sufficient condition for the solvability of a rank one differential (resp. $q$-difference) equation over the Amices ring. We also extend to that ring a Birkoff decomposition result, originally due to Motzkin.

Small Connections are cyclic

The main local invariants of a (one variable) differential module over the complex numbers are given by means of a cyclic basis. In the $p$-adic setting the existence of a cyclic vector is often unknown. We investigate the existence of such a cyclic vector in a Banach algebra. We follow the explicit method of Katz, and we prove the existence of such a cyclic vector under the assumption that the matrix of the derivation is small enough in norm.

The convergence Newton polygon of a $p$-adic differential equation IV : local and global index theorems

Let $X$ be a quasi-smooth Berkovich curve over a field of characteristic $0$ and let $mathscr{F}$ be a locally free $mathscr{O}_{X}$-module with connection. In this paper, we prove local and global criteria to ensure the finite-dimensionality of the de Rham cohomology of $mathscr{F}$. Moreover, we state a global Grothendieck-Ogg-Shafarevich formula that relates the index of $mathscr{F}$ in the sense of de Rham cohomology to the Euler characteristic of $X$ and expresses the difference as a sum of irregularities. We also derive super-harmonicity results for the partial heights of the convergence Newton polygon of $mathscr{F}$.

The convergence Newton polygon of a $p$-adic differential equation III : global decomposition and controlling graphs

We deal with locally free $mathcal{O}_X$-modules with connection over a Berkovich curve $X$. As a main result we prove local and global decomposition theorems of such objects by the radii of convergence of their solutions. We also derive a bound of the number of edges of the controlling graph, in terms of the geometry of the curve and the rank of the equation. As an application we provide a classification result of such equations over elliptic curves.

An algorithm computing non solvable spectral radii of $p$-adic differential equations

We obtain an algorithm computing explicitly the values of the non solvable spectral radii of convergence of the solutions of a differential module over a point of type 2, 3 or 4 of the Berkovich affine line.

Continuity and finiteness of the radius of convergence of a p-adic differential equation via potential theory

We study the radius of convergence of a differential equation on a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Several properties of this function are known: F. Baldassarri proved that it is continuous and the authors showed that it factorizes by the retraction through a locally finite graph. Here, assuming that the curve has no boundary or that the differential equation is overconvergent, we provide a shorter proof of both results by using potential theory on Berkovich curves.

The convergence Newton polygon of a p-adic differential equation II: Continuity and finiteness on Berkovich curves

We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorize by the retraction through a locally finite subgraph of the curve.

The convergence Newton polygon of a $p$-adic differential equation I : Affinoid domains of the Berkovich affine line

We prove that the radii of convergence of the solutions of a $p$-adic differential equation $mathcal{F}$ over an affinoid domain $X$ of the Berkovich affine line are continuous functions on $X$ that factorize through the retraction of $XtoGamma$ of $X$ onto a finite graph $Gammasubseteq X$. We also prove their super-harmonicity properties. Roughly speaking, this finiteness result means that the behavior of the radii as functions on $X$ is controlled by a finite family of data.

Arithmetic and Differential Swan Conductors of rank one representations with finite local monodromy

We consider a complete discrete valuation field of characteristic p, with possibly non perfect residue field. Let V be a rank one continuous representation with finite local monodromy of its absolute Galois group. We will prove that the Arithmetic Swan conductor of V (defined after Kato in [Kat89] which fits in the more general theory of [AS02] and [AS06]) coincides with the Differential Swan conductor of the associated differential module $D^{dag}(V)$ defined by Kedlaya in [Ked]. This construction is a generalization to the non perfect residue case of the Fontaines formalism as presented in [Tsu98a]. Our method of proof will allow us to give a new interpretation of the Refined Swan Conductor.