2007/09/13 by Francesco Baldassarri, Baldassarri, Francesco, Lucia Di Vizio +1 · 1 citation
Mathematics · #12H25 #14G22 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories #math.AG #math.NT #msc:12H25 #msc:14G22
paper · pdf · doi:10.48550/arxiv.0709.2008
19 pages. We have simplified and improved the exposition
openalex publication_date 2007/09/13 · arxiv created 2008/03/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a vector bundle with integrable connection (\cE,\na) on an analytic domain U in the generic fiber \cXη of a smooth formal p-adic scheme \cX, in the sense of Berkovich. We define the diameter δ\cX(ξ,U) of U at ξ∈ U, the radius ρ\cX(ξ) of the point ξ∈\cXη, the radius of convergence of solutions of (\cE,\na) at ξ, R(ξ) = R\cX(ξ, U,(\cE, \na)). We discuss (semi-) continuity of these functions with respect to the Berkovich topology. In particular, under we prove under certain assumptions that δ\cX(ξ,U), ρ\cX(ξ) and Rξ(U,\cE,\na) are upper semicontinuous functions of ξ; for Laurent domains in the affine space, δ\cX(-,U) is continuous. In the classical case of an affinoid domain U of the analytic affine line, R is a continuous function.