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The different for base change of arithmetic curves

Art Waeterschoot

TL;DR

The paper develops a framework to study reduction types of arithmetic curves under wild base change by constructing simultaneous skeleta for covers of Berkovich analytifications and applying a skeletal Riemann-Hurwitz formula. It introduces a combinatorial slope theorem that links ramification data to dual-graph invariants and provides new proofs of results by Lorenzini and Obus–Wewers, as well as resolving Lorenzini's question on Euler characteristics of resolution graphs for $p$-cyclic arithmetic surface quotient singularities. The approach hinges on the different function $oldsymbol{oldsymbol{ extdelta}}$ and a base-change perspective on skeleta, enabling a unified treatment of elliptic curve reductions and wild quotient singularities. Overall, the work offers a versatile toolkit for translating ramification and base-change phenomena into combinatorial data on skeletal graphs, with potential connections to McKay-type correspondences and broader arithmetic-geometry applications.

Abstract

We introduce a method for studying reduction types of arithmetic curves and wildly ramified base change. We give new proofs of earlier results of Lorenzini and Obus-Wewers, and resolve a question of Lorenzini on the Euler characteristic of the resolution graph of a $p$-cyclic arithmetic surface quotient singularity. Our method consists of constructing a simultaneous skeleton for the associated cover of Berkovich analytifications and applying a skeletal Riemann-Hurwitz formula.

The different for base change of arithmetic curves

TL;DR

The paper develops a framework to study reduction types of arithmetic curves under wild base change by constructing simultaneous skeleta for covers of Berkovich analytifications and applying a skeletal Riemann-Hurwitz formula. It introduces a combinatorial slope theorem that links ramification data to dual-graph invariants and provides new proofs of results by Lorenzini and Obus–Wewers, as well as resolving Lorenzini's question on Euler characteristics of resolution graphs for -cyclic arithmetic surface quotient singularities. The approach hinges on the different function and a base-change perspective on skeleta, enabling a unified treatment of elliptic curve reductions and wild quotient singularities. Overall, the work offers a versatile toolkit for translating ramification and base-change phenomena into combinatorial data on skeletal graphs, with potential connections to McKay-type correspondences and broader arithmetic-geometry applications.

Abstract

We introduce a method for studying reduction types of arithmetic curves and wildly ramified base change. We give new proofs of earlier results of Lorenzini and Obus-Wewers, and resolve a question of Lorenzini on the Euler characteristic of the resolution graph of a -cyclic arithmetic surface quotient singularity. Our method consists of constructing a simultaneous skeleton for the associated cover of Berkovich analytifications and applying a skeletal Riemann-Hurwitz formula.

Paper Structure

This paper contains 4 sections, 7 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on differents of discretely valued fields
  3. Models and skeleta of curves
  4. Skeleta and base change

Key Result

Theorem 1.1

Suppose that $\Gamma$ is a subgraph of $C^{\mathrm{an}}$ which contains a skeleton. Then $\Gamma$ is a skeleton if and only if $\chi(\Gamma)=\chi(C)$.

Figures (2)

  • Figure 1.5: Picture of a simultaneous skeleton $\Gamma'\to\Gamma$ for Example \ref{['example pot mult']}. The markings are the multiplicities of the vertices. The different $\delta$ is constant on the coloured part of $\Gamma$ and varies linearly with slope $2$ from $\delta=0$ on the loop to $\delta=1/2$ above $\Gamma^{\mathrm{t}}$
  • Figure 1.8: Resolution graph $\Gamma_Q$ of a weakly wild $p$-cyclic arithmetic surface quotient singularities; we have $j_Q=1$ and $\chi(\Gamma_Q)=2-p$. The markings indicate multiplicities and $r$ is an integer such that $0<r<p$.

Theorems & Definitions (28)

  • Theorem 1.1: Enlarging skeleta
  • Remark 1.2
  • Theorem 1.3: Potentially multiplicative elliptic curves
  • Example 1.4
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.9: Outlook
  • Definition 2.1
  • Proposition 2.2: Properties of the different
  • proof
  • ...and 18 more