Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Triangulations of non-archimedean curves, semi-stable reduction, and
  ramification   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.

10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
29
The categories of this paper are: funct-an