Corpus ID: 119301158

Models of curves over DVRs

@article{Dokchitser2018ModelsOC,
title={Models of curves over DVRs},
author={Tim Dokchitser},
journal={arXiv: Number Theory},
year={2018}
}
• T. Dokchitser
• Published 29 June 2018
• Mathematics
• arXiv: Number Theory
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.

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References

SHOWING 1-10 OF 54 REFERENCES
Arithmetic of hyperelliptic curves over local fields
• Mathematics
• 2017
We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a "cluster picture" associated to the curve, that describes the p-adic distancesExpand
Euler factors determine local Weil representations
• Mathematics
• 2011
We show that a Frobenius-semisimple Weil representation over a local fieldK is determined by its Euler factors over the extensions ofK. The construction is explicit, and we illustrate it for l-adicExpand
Quotients of hyperelliptic curves and etale cohomology
• Mathematics
• 2015
We study hyperelliptic curves C with an action of an affine group of automorphisms G. We establish a closed form expression for the quotient curve C/G and for the first etale cohomology group of C asExpand
COMPUTING L-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES
• Mathematics
• Glasgow Mathematical Journal
• 2016
Abstract We give an explicit description of the stable reduction of superelliptic curves of the form y n =f(x) at primes $\mathfrak{p}$ whose residue characteristic is prime to the exponent n. WeExpand
Stable reduction of curves and tame ramification
We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuationExpand
Weights of exponential sums, intersection cohomology, and Newton polyhedra
• Mathematics
• 1991
(1.1) Throughout this paper k always denotes a finite field Fq with q elements, and E a prime number not dividing q. The algebraic closure of a field K is denoted by / ( . Let ~b: k--+ C • be aExpand
Sur le défaut de semi-stabilité des courbes elliptiques à réduction additive
AbstractLet K be a field of characteristics 0 complete with respect to a discrete valuation v, with a perfect residue field of characteristic p>0. Let $$\vec K$$ be an algebraic closure of K and KnrExpand
Models of Curves and Wild Ramification
Let K be a complete discrete valuation field with ring of integers OK and residue field k of characteristic p ≥ 0, assumed to be algebraically closed. Let X/K denote a smooth proper geometricallyExpand
Elliptic fibers over non-perfect residue fields
Abstract Kodaira and Neron classified and described the geometry of the special fibers of the Neron model of an elliptic curve defined over a discrete valuation ring with a perfect residue field.Expand
On nondegeneracy of curves
• Mathematics
• 2008
We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 overExpand