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Riemann-Hurwitz Formula for Arithmetic Surfaces

Ziyang Zhu

TL;DR

The paper extends the classical Riemann-Hurwitz framework to arithmetic surfaces by factoring finite morphisms through base-change arithmetic surfaces and separating ramification into geometric (horizontal) and arithmetic (vertical) components. It derives a divisor-level RH formula K_{X/ Z} = φ^*K_{Y/ Z} + R_geo + R_ari, clarifying how each ramification contribution arises from the morphism’s factorization and base-change properties. The work connects these geometric/arithmetic ramification terms to Grothendieck-Riemann-Roch and, in the Arakelov setting, to arithmetic Chow groups, providing both a concrete example and pathways to Arakelov RH formulas. Overall, it furnishes a structured framework to compute and interpret ramification on arithmetic surfaces and to extend RH-type results into Arakelov theory. A key illustrative example computes the ramification data for a concrete Z-based morphism, illuminating the interaction between horizontal and vertical ramification in practice.

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.

Riemann-Hurwitz Formula for Arithmetic Surfaces

TL;DR

The paper extends the classical Riemann-Hurwitz framework to arithmetic surfaces by factoring finite morphisms through base-change arithmetic surfaces and separating ramification into geometric (horizontal) and arithmetic (vertical) components. It derives a divisor-level RH formula K_{X/ Z} = φ^*K_{Y/ Z} + R_geo + R_ari, clarifying how each ramification contribution arises from the morphism’s factorization and base-change properties. The work connects these geometric/arithmetic ramification terms to Grothendieck-Riemann-Roch and, in the Arakelov setting, to arithmetic Chow groups, providing both a concrete example and pathways to Arakelov RH formulas. Overall, it furnishes a structured framework to compute and interpret ramification on arithmetic surfaces and to extend RH-type results into Arakelov theory. A key illustrative example computes the ramification data for a concrete Z-based morphism, illuminating the interaction between horizontal and vertical ramification in practice.

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.

Paper Structure

This paper contains 6 sections, 15 theorems, 52 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Geometry of Arithmetic Surfaces
  3. Factorization of Morphisms
  4. Ramification Theory: Geometric v.s. Arithmetic
  5. The Arithmetic Riemann-Hurwitz Formula
  6. The Grothendieck-Riemann-Roch Theorem

Key Result

Proposition 2

Let $A$ be a unique factorization domain, $A[\theta]$ is a single ring extension with $\theta$ a root of some monic polynomial $f\in A[x]$ such that $A[\theta]$ is regular. Consider the discriminate of $f$, where $\theta_i$ are roots of $f$. Then the ramification divisor of $\mathop{}\!\mathrm{Spec}(A[\theta])\to\mathop{}\!\mathrm{Spec}(A)$ is generated by some prime divisors of $D(f)$.

Theorems & Definitions (25)

  • Definition 1
  • Proposition 2
  • Example 3
  • Definition 4
  • Theorem 5
  • Proposition 6
  • Example 7
  • Lemma 8
  • Proposition 9
  • Example 10
  • ...and 15 more