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Register a new userContinuity and finiteness of the radius of convergence of a p-adic differential equation via potential theory
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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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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.
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.
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 $mathscr{F}$.
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.
Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p is greater than or equal to d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for primes between d/2 and d. We also explore a one-parameter family of cubic polynomials over the 2-adic numbers to illustrate that the p-adic Mandelbrot set can be quite complicated when p is less than d, in contrast with the simple and well-understood p > d case.
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