Convergence of solutions of $p$-adic differential equations

Given an ordinary differential equation whose coefficients are
meromorphic functions of a complex variable, the only obstruction to
convergence of local solutions in a disc is the presence of
singularities within the disc. It was observed decades ago that this
fails if one replaces “complex by “$p$-adic, e.g., consider the
exponential function. In recent work of Baldassarri, Poineau, Pulita,
and the speaker, it has emerged that the convergence properties of such
solutions in the $p$-adic case can be described quite simply in terms of
Berkovich analytic geometry. We will give this description (without
assuming any prior familiarity with Berkovich’s theory) and mention some
applications to studying wild ramification of covers of $p$-adic curves.