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Register a new userThe convergence Newton polygon of a $p$-adic differential equation IV : local and global index theorems
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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 $mathscr{F}$.
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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.
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 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.
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.
Let F be a function field in one variable over a p-adic field and D a central division algebra over F of degree n coprime to p. We prove that Suslin invariant detects whether an element in F is a reduced norm. This leads to a local-global principle for reduced norms with respect to all discrete valuations of F.
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