# Continuity of the radius of convergence of p-adic differential equations on Berkovich analytic spaces - Mathematics > Number Theory

Abstract: We consider a vector bundle with integrable connection \cE, a on ananalytic domain U in the generic fiber \cX {\eta} of a smooth formal p-adicscheme \cX, in the sense of Berkovich. We define the \emph{diameter}\delta {\cX}\xi,U of U at \xi\in U, the \emph{radius} ho {\cX}\xi of thepoint \xi\in\cX {\eta}, the \emph{radius of convergence} of solutions of\cE, a at \xi, R\xi = R {\cX}\xi, U,\cE, a. We discuss semi-continuity of these functions with respect to the Berkovich topology. Inparticular, under we prove under certain assumptions that \delta {\cX}\xi,U, ho {\cX}\xi and R {\xi}U,\cE, a are upper semicontinuous functions of\xi; for Laurent domains in the affine space, \delta {\cX}-,U is continuous.In the classical case of an affinoid domain U of the analytic affine line, R isa continuous function.

Author: Francesco Baldassarri, Lucia Di Vizio

Source: https://arxiv.org/