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

This paper deals with connections on non-archimedean, especially 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 a geometric, intrinsic notion of normalized radius of convergence of a full set of local solutions as a function on the curve, with values in (0, 1]. For a sufficiently refined choice of the semistable model, we prove continuity, logarithmic concavity and logarithmic piece-wise linearity of that function. We introduce and characterize Robba connections, that is connections whose sheaf of solutions is constant on any open disk contained in the curve, precisely as it happens in the classical case.