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Triangulations of non-archimedean curves, semi-stable reduction, and ramification

Lorenzo Fantini, Daniele Turchetti

TL;DR

The paper develops a non-archimedean analytic framework for understanding semi-stable reduction of curves via minimal triangulations of Berkovich analytifications. It proves that the least common multiple of the multiplicities at the minimal triangulation, $\mathrm{lcm}\{ m(x) \mid x \in V_{\rm min-tr} \}$, divides the degree of the minimal Galois extension $[L:K]$, and, when the residue characteristic $p$ does not divide this value, that the degree equals the lcm, providing a new proof of a classical Saito-type criterion. The authors extend the approach to marked curves and give explicit tame triangulations for elliptic curves, including a detailed reduction-type description, while also illustrating how wild ramification can invalidate the tame picture. The work highlights both the power and the limitations of minimal triangulations in encoding ramification data and paves the way for further understanding of wild ramification via Berkovich geometry.

Abstract

Let $K$ be a complete discretely valued field with algebraically closed residue field and let $\mathfrak C$ be a smooth projective and geometrically connected algebraic $K$-curve of genus $g$. Assume that $g\geq 2$, so that there exists a minimal finite Galois extension $L$ of $K$ such that $\mathfrak C_L$ admits a semi-stable model. In this paper, we study the extension $L|K$ in terms of the \emph{minimal triangulation} of $C$, a distinguished finite subset of the Berkovich analytification $C$ of $\mathfrak C$. We prove that the least common multiple $d$ of the multiplicities of the points of the minimal triangulation always divides the degree $[L:K]$. Moreover, if $d$ is prime to the residue characteristic of $K$, then we show that $d=[L:K]$, obtaining a new proof of a classical theorem of T. Saito. We then discuss curves with marked points, which allows us to prove analogous results in the case of elliptic curves, whose minimal triangulations we describe in full in the tame case. In the last section, we illustrate through several examples how our results explain the failure of the most natural extensions of Saito's theorem to the wildly ramified case.

Triangulations of non-archimedean curves, semi-stable reduction, and ramification

TL;DR

The paper develops a non-archimedean analytic framework for understanding semi-stable reduction of curves via minimal triangulations of Berkovich analytifications. It proves that the least common multiple of the multiplicities at the minimal triangulation, , divides the degree of the minimal Galois extension , and, when the residue characteristic does not divide this value, that the degree equals the lcm, providing a new proof of a classical Saito-type criterion. The authors extend the approach to marked curves and give explicit tame triangulations for elliptic curves, including a detailed reduction-type description, while also illustrating how wild ramification can invalidate the tame picture. The work highlights both the power and the limitations of minimal triangulations in encoding ramification data and paves the way for further understanding of wild ramification via Berkovich geometry.

Abstract

Let be a complete discretely valued field with algebraically closed residue field and let be a smooth projective and geometrically connected algebraic -curve of genus . Assume that , so that there exists a minimal finite Galois extension of such that admits a semi-stable model. In this paper, we study the extension in terms of the \emph{minimal triangulation} of , a distinguished finite subset of the Berkovich analytification of . We prove that the least common multiple of the multiplicities of the points of the minimal triangulation always divides the degree . Moreover, if is prime to the residue characteristic of , then we show that , obtaining a new proof of a classical theorem of T. Saito. We then discuss curves with marked points, which allows us to prove analogous results in the case of elliptic curves, whose minimal triangulations we describe in full in the tame case. In the last section, we illustrate through several examples how our results explain the failure of the most natural extensions of Saito's theorem to the wildly ramified case.

Paper Structure

This paper contains 10 sections, 14 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Non-archimedean curves
  3. Virtual discs and virtual annuli
  4. Structure of analytic curves and triangulations
  5. Minimal triangulations and ramification
  6. Snc models and vertex sets
  7. The tamely ramified case
  8. Marked curves and tame triangulations of elliptic curves
  9. First steps in wildlife observation
  10. Open fractional annuli and regularity

Key Result

Proposition 2.4

Let $L$ be a finite, separable extension of $K$ of degree $d$, let $\tau\colon X_{L}\to X$ be the base change morphism, let $y$ be a type 2 point of $X_{L}$ and set $x=\tau(y)$. Then $m(x) \geqslant m(y)$ and $\frac{m(x)}{\gcd(m(x),d)}$ divides $m(y)$. If $L|K$ is tamely ramified or $p$ does not div

Figures (5)

  • Figure 1: A disc, in blue, and the annulus $A_{2,K}$, in red, seen as subspaces of ${\mathbb{A}^{1,\mathrm{an}}_K}$.
  • Figure 2: The bottom half of the figure depicts the virtual disc of Example \ref{['example:virtual_several_fractional_discs']} with $m=2$ (on the left) and the virtual annulus with one end of Example \ref{['example_virtual_annulus_oneend']} (on the right). They are both trivialized by the base change to $K(\sqrt\pi)$, as shown in the top half of the figure. The numbers between parentheses denote the multiplicities of the nearby type 2 points, and the skeletons of the virtual annulus and of its trivialization are depicted in bold.
  • Figure 3:
  • Figure 4:
  • Figure 5:

Theorems & Definitions (37)

  • Proposition 2.4
  • proof
  • Example 3.12
  • proof
  • Proposition 3.15
  • proof
  • Theorem 4.3
  • Remark 4.5
  • Remark 4.6
  • Proposition 4.10
  • ...and 27 more