Riemann-Hurwitz Formula for Arithmetic Surfaces

Abstract

In this paper, we presents a method for factoring morphisms between arithmetic surfaces based on the regularity of arithmetic surfaces. Using this factorization, we derive a Riemann-Hurwitz formula satisfied by the ramification divisor and the canonical divisor on arithmetic surfaces. We also extend this formula to Arakelov theory.

14 pages, we have corrected some errors from the previous version, comments welcome!