Wild ramification of p-adic analytic curves

Wild ramification is commonly regarded as pathological. Contrary to this belief, we will explain how it can be effectively controlled via p-adic analytic geometry. Let f:C->D be a cover of p-adic curves. By studying the p-adic analysis of the pullback map f*:Omega_D -> Omega_C on differential forms, one can obtain a ‘different function’ delta: C^{an} ->\R_{ge 0} that measures ramification of the associated p-adic analytification f^{an}:C^{an}\to D^{an}. The potential theory of the different reveals new `analytic’ Riemann-Hurwitz formula: for a simultaneous skeleton the Laplacian of the different equals the `tropical’ relative canonical divisor. This complements results of Temkin e.a. over an algebraically closed ground field, extending them to discretely valued fields. The p-adic different function provides a new tool to study surfaces fibered over Z_p. We illustrate this by explaining the proof of a conjecture of Lorenzini on surface p-cyclic quotient singularities, which relates the combinatorial complexity of a resolution to the valuation theory of local rings.