# Models of curves over DVRs

@article{Dokchitser2018ModelsOC, title={Models of curves over DVRs}, author={Tim Dokchitser}, journal={arXiv: Number Theory}, year={2018} }

Let C be a smooth projective curve over a discretely valued field K, defined by an affine equation f(x,y)=0. We construct a model of C over the ring of integers of K using a toroidal embedding associated to the Newton polygon of f. We show that under 'generic' conditions it is regular with normal crossings, and determine when it is minimal, the global sections of its relative dualising sheaf, and the tame part of the first etale cohomology of C.

#### 13 Citations

Models of Bihyperelliptic Curves

- Mathematics
- 2021

We give an explicit description of the minimal regular model of bihyperelliptic curves with semistable reduction over a local field of odd residue characteristic. We do this using a generalisation of… Expand

On Galois representations of superelliptic curves

- Mathematics
- 2020

A superelliptic curve over a DVR O of residual characteristic p is a curve given by an equation C : yn = f(x). The purpose of the present article is to describe the Galois representation attached to… Expand

Integral differential forms for superelliptic curves

- Mathematics
- 2020

Given a superelliptic curve $Y_K : y^n = f(x)$ over a local field $K$, we describe the theoretical background and an implementation of a new algorithm for computing the $\mathcal{O}_K$-lattice of… Expand

Models of hyperelliptic curves with tame potentially semistable reduction

- Mathematics
- 2019

Let $C$ be a hyperelliptic curve $y^2 = f(x)$ over a discretely valued field $K$. The $p$-adic distances between the roots of $f(x)$ can be described by a completely combinatorial object known as the… Expand

Models and Integral Differentials of Hyperelliptic Curves

- Mathematics
- 2020

Let $C:y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 2$, defined over a discretely valued complete field $K$, with ring of integers $O_K$. Under certain conditions on $C$, mild when residue… Expand

Superelliptic curves with large Galois images.

- Mathematics
- 2020

Let r>2 and l be primes. In this paper we study the mod l Galois representations attached to curves of the form y^r = f(x) where f is monic and has coefficients belonging to the r-th cyclotomic… Expand

A user's guide to the local arithmetic of hyperelliptic curves

- Mathematics
- 2020

A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of… Expand

A generalization of the Newton-Puiseux algorithm for semistable models

- Mathematics
- 2020

In this paper we give an algorithm that calculates a skeleton of a tame covering of curves over a complete discretely valued field. The algorithm mainly relies on the {tame simultaneous semistable… Expand

Root number of the Jacobian of $y^2=x^p+a$

- Mathematics
- 2021

Let C/Q be a hyperelliptic curve with an affine model of the form y = x + a. We explicitly determine the root number of the Jacobian of C, with particular focus on the local root number at p where C… Expand

Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic.

- Mathematics
- 2019

We prove a conductor-discriminant inequality for all hyperelliptic curves defined over discretely valued fields $K$ with perfect residue field of characteristic not $2$. Specifically, if such a curve… Expand

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