English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $K$ be a complete discretely valued field with algebraically closedresidue field and let $\mathfrak C$ be a smooth projective and geometricallyconnected algebraic $K$-curve of genus $g$. Assume that $g\geq 2$, so thatthere exists a minimal finite Galois extension $L$ of $K$ such that $\mathfrakC_L$ admits a semi-stable model. In this paper, we study the extension $L|K$ interms of the \emph{minimal triangulation} of $C$, a distinguished finite subsetof the Berkovich analytification $C$ of $\mathfrak C$. We prove that the leastcommon multiple $d$ of the multiplicities of the points of the minimaltriangulation always divides the degree $[L:K]$. Moreover, if $d$ is prime tothe residue characteristic of $K$, then we show that $d=[L:K]$, obtaining a newproof of a classical theorem of T. Saito. We then discuss curves with markedpoints, which allows us to prove analogous results in the case of ellipticcurves, whose minimal triangulations we describe in full in the tame case. Inthe last section, we illustrate through several examples how our resultsexplain the failure of the most natural extensions of Saito's theorem to thewildly ramified case.

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