Harmonische overdekkingen van skeletten en wild vertakte krommen

Wild ramification is a phenomenon of algebraic geometry in positive residue characteristic p where intuition often breaks down. Roughly speaking it happens when derivatives of polynomials are divisible by too many factors of p. Contrary to the common belief that wild ramification is pathological behaviour, in some cases it can be controlled effectively via nonarchimedean geometry. Nonarchimedean geometry is the theory of analytic spaces over valued fields like the field of p-adic numbers Q_p. In the 1980s Berkovich discovered a natural formulation of such a theory in which nonarchimedean analytic spaces are built from piecewise linear subspaces called skeleta. In this thesis we measure the wild ramification for maps from one skeleton to another and give applications to arithmetic curves and base change. Our proof techniques use ideas from logarithmic geometry, tropical geometry and potential theory. More precisely, let f:X->Y be a finite seperable cover of proper smooth varieties over a complete discretely valued field k and let Sigma_X->Sigma_Y be a simultaneous skeleton of the Berkovich analytification f^an:X^an->Y^an. Then complementing earlier work of Temkin e.a. we construct a real-valued 'different' function delta_f:Sigma_X ->R_≥0 measuring wild ramification along Sigma_X->Sigma_Y. Our central result shows that the different delta_f is a piecewise linear function with integral slopes satisfying a ''Riemann-Hurwitz formula'' Δ(delta_f)=K_{Sigma_X/Sigma_Y} where Δ is the Laplacian and K_{\Sigma_X/\Sigma_Y} denotes the tropical canonical divisor. In the applications we give new proofs of Theorems of Lorenzini and Obus-Wewers on modulop reduction types of elliptic curves and ordinary curves, and we affirm a question of Lorenzini on 2-dimensional arithmetic Z/pZ-Galois quotient singularities by showing that the Euler characteristic of the nearby fiber of such a singularity can be computed from the valuation theory of local rings.