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Andrea Pulita

Publications and source records attributed to Andrea Pulita.

15 recordsLinked to original sources
arXiv2024

The convergence Newton polygon of a $p$-adic differential equation IV : controlling graphs

In our previous works we proved a finiteness property of the radii of convergence functions associated with a vector bundle with connection on $p$-adic analytic curves. We showed that the radii are locally constant functions outside a locally finite graph in the curve, called controlling graph. In this paper we refine that finiteness results by giving a bound on the size of the controlling graph in terms of the geometry of the curve and the rank of the module. This is based on super-harmonicity properties of radii of convergence and partial heights of the Newton polygon. Under suitable assumptions, we relate the size of the controlling graph associated with the total height of the convergence Newton polygon to the Euler characteristic in the sense of de Rham cohomology.

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arXiv2024

Regular singular differential modules over differential rings

We obtain Fuchs decomposition theorem for regular singular differential modules over a large class of differential rings. We provide a definition of regularity inspired by differential Galois theory and we deduce the classical equivalence with vector spaces endowed with an automorphism.

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arXiv2024

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

In this paper and its sequel we consider locally-free $\mathscr{O}_X$-modules together with a connection over a quasi-smooth Berkovich curve $X$. We obtain necessary and sufficient conditions for the finite dimensionality of their de Rham cohomology over local domains such as disks and annuli. We deal with both analytic and meromorphic connections and we derive index formulas relating the index to the behavior of the radii of convergence of their solutions at the boundary of the curve $X$. We introduce the notion of absolute local index. We prove that it is an intrinsic notion extending the previous notions of Robba's generalized index and that of $p$-adic exponents. This condition arises at the boundary of the curve $X$ and it is an exact condition for the finite dimensionality of the de Rham cohomology. We derive comparison results between formal, meromorphic and analytic de Rham cohomologies.

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arXiv2024

The convergence Newton polygon of a $p$-adic differential equation III : global decompositions

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.

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arXiv2021

An uncountable Mittag-Leffler condition with an application to ultrametric locally convex vector spaces

Mittag-Leffler condition ensures the exactness of the inverse limit of short exact sequences indexed on a partially ordered set $(I,\leq)$ admitting a $countable$ cofinal subset. We extend Mittag-Leffler condition by relatively relaxing the countability assumption. As an application we prove an ultrametric analogous of a result of V.P.Palamodov in relation with the acyclicity of Frechet spaces with respect to the completion functor.

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arXiv2019

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.

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arXiv2016

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 $σ$ of $X$, and if the automorphism is sufficiently close to the identity, then $\mathscr{F}$ acquires a semi-linear action of $σ$ (i.e. lifting that on $X$). This generalizes the previous works of Yves André, Lucia Di Vizio, and the author about $p$-adic $q$-difference equations. We also obtain an application to Morita's $p$-adic Gamma function, and to related values of $p$-adic $L$-functions.

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arXiv2014

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.

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arXiv2014

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 $X\toΓ$ of $X$ onto a finite graph $Γ\subseteq X$. 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.

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arXiv2012

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.

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arXiv2008

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 $D^†(V)$ defined by Kedlaya in [Ked]. This construction is a generalization to the non perfect residue case of the Fontaine's formalism as presented in [Tsu98a]. Our method of proof will allow us to give a new interpretation of the Refined Swan Conductor.

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arXiv2007

\bs{p}-Adic Confluence of $\bs{q}$-Difference Equations

We develop the theory of $p$-adic confluence of $q$-difference equations. The main result is the surprising fact that, in the $p$-adic framework, a function is solution of a differential equation if and only if it is solution of a $q$-difference equation. This fact implies an equivalence, called ``Confluence'', between the category of differential equations and those of $q$-difference equations. We obtain this result by introducing a category of ``sheaves'' on the disk $\mathrm{D}^-(1,1)$, whose stalk at 1 is a differential equation, the stalk at $q$ is a $q$-difference equation if $q$ is not a root of unity $ξ$, and the stalk at a root of unity is a mixed object, formed by a differential equation and an action of $σ_ξ$.

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arXiv2006

Rank One Solvable p-adic Differential Equation and Finite Abelian Characters via Lubin-Tate groups

We introduce a new class of exponentials of Artin-Hasse type, called $\boldsymbolπ$-exponentials. These exponentials depends on the choice of a generator $\boldsymbolπ$ of the Tate module of a Lubin-Tate group $\mathfrak{G}$ over $\mathbb{Z}_p$. They arise naturally as solutions of solvable differential modules over the Robba ring. If $\mathfrak{G}$ is isomorphic to $\hat{\mathbb{G}}_m$ over $\mathbb{Z}_p$, we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation of the Artin-Schreier complex into the Kummer complex. This provides an explicit generator of the Kummer unramified extension of $\mathcal{E}^†_{K_{\infty}}$, whose residue field is a given Artin-Schreier extension of k((t)), where k is the residue field of K. We then compute explicitely the group, under tensor product, of isomorphism classes of rank one solvable differential equations. Moreover, we get a canonical way to compute the rank one $ϕ$-module over $\mathcal{E}^†_{K_{\infty}}$ attached to a rank one representation of $Gal(k((t))^{sep}/k((t)))$, defined by an Artin-Schreier character.

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