Continuity of the radius of convergence of differential equations on $p$-adic analytic curves

Details Continuity of the radius of convergence of differential equations on $p$-adic analytic curves Continuity of the radius of convergence of differential equations on $p$-adic analytic curves

2008-09-15

arxiv.org/abs/0809.2479v6

Mathematics

Algebraic Geometry

51 pages, 4 figures

Scientific paper

This paper deals with connections on $p$-adic analytic curves, in the sense of Berkovich. The curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them. For any choice of a semistable formal model of the curve, we define an intrinsic notion of normalized radius of convergence as a function on the curve, with values in $(0,1]$. For a sufficiently refined choice of the semistable model, we prove continuity and logarithmic concavity of that function. We characterize \emph{Robba connections}, that is connections whose sheaf of solutions is constant on any open disk contained in the curve.

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