TY - JOUR

T1 - Morphisms of Berkovich curves and the different function

AU - Cohen, Adina

AU - Temkin, Michael

AU - Trushin, Dmitri

N1 - Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/11/5

Y1 - 2016/11/5

N2 - Given a generically étale morphism f:Y→X of quasi-smooth Berkovich curves, we define a different function δf:Y→[0,1] that measures the wildness of the topological ramification locus of f. This provides a new invariant for studying f, which cannot be obtained by the usual reduction techniques. We prove that δf is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann–Hurwitz formula, and show that δf can be used to explicitly construct the simultaneous skeletons of X and Y. As another application, we use our results to completely describe the topological ramification locus of f when its degree equals to the residue characteristic p.

AB - Given a generically étale morphism f:Y→X of quasi-smooth Berkovich curves, we define a different function δf:Y→[0,1] that measures the wildness of the topological ramification locus of f. This provides a new invariant for studying f, which cannot be obtained by the usual reduction techniques. We prove that δf is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann–Hurwitz formula, and show that δf can be used to explicitly construct the simultaneous skeletons of X and Y. As another application, we use our results to completely describe the topological ramification locus of f when its degree equals to the residue characteristic p.

KW - Berkovich analytic spaces

KW - The different

KW - Topological ramification

UR - http://www.scopus.com/inward/record.url?scp=84984605160&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2016.08.029

DO - 10.1016/j.aim.2016.08.029

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AN - SCOPUS:84984605160

SN - 0001-8708

VL - 303

SP - 800

EP - 858

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -