Three point covers with bad reduction

Bad reduction of genus $3$ curves with complex multiplication

Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $\mathfrak{p}$ of $M$ such that the stable reduction of $C$ at $\mathfrak{p}$ contains three irreducible components of genus $1$.

The Chabauty-Coleman bound at a prime of bad reduction and clifford bounds for geometric rank functions

Let X be a curve over a number field K with genus g>=2, $\pp$ a prime of O_K over an unramified rational prime p>2r, J the Jacobian of X, r=rank J(K), and $\scrX$ a regular proper model of X at $\pp$. Suppose r<g. We prove that #X(K)<=#\scrX(F_{\pp})+2r, extending the refined version of the Chabauty-Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Clifford's theorem.

Lame curves with bad reduction

Lame curves are a particular class of elliptic curves (with a torsion point attached to them) which naturally arise when studying Lame operators with finite monodromy. They can be realized as covers of the projective line unramified outside three points and can be defined over number fields. This paper investigates their p-adic properties. The main ingredient is formal/rigid geometry and in particular the use of p-adic theta functions. As a consequence, we can completely enumerate Lamé curves with bad reduction (by giving, among the others, the p-adic valuation of their j-invariant) and descri…

具有坏约化的K3曲面的Picard秩跃变

Picard rank jumps for K3 surfaces with bad reduction

该研究将K3曲面Picard秩跃变的相关结果推广到数域上具潜在坏约化的K3曲面及有限域代数闭包上曲线的一般平凡非同平凡K3曲面族，并给出正交和酉Shimura簇平凡Hecke轨道猜想的新证明。

Projective varieties with bad reduction at 3 only

Suppose F=W(k)[1/p] where W(k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p>2. We construct integral theory of p-adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0,p). This modification of Breuil's theory results in the following application in the spirit of Shafarevich's Conjecture. If Y is a projective algebraic variety over the field of rational numbers with good reduction modulo all primes different from 3 and semi-stable reduction modulo 3 then for the Hodge numbers of the complexification…

Approximation at places of bad reduction

This paper addresses weak approximation for rationally connected varieties defined over the function field of a curve, especially at places of bad reduction. Our approach entails analyzing the rational connectivity of the smooth locus of singular reductions of the variety. As an application, we prove weak approximation for cubic surfaces with square-free discriminant.

具有复乘的亏格二曲线的坏约化

Bad Reduction of Genus Two Curves with complex multiplication

本文研究具有双二次CM域复乘的亏格二曲线，通过正交Shimura簇上的CM环显式描述其雅可比簇，推导稳定坏约化素数的上界并证明其存在性。

Minimal number of points with bad reduction for elliptic curves over P^1

In this work we use elementary methods to discuss the question of the minimal number of points with bad reduction over the projective line for elliptic curves E/k(T) which are non-constant resp. have non-constant j-invariant.

Bad reduction of the Brauer-Manin obstruction

We relate the Brauer group of a smooth variety over a p-adic field to the geometry of the special fibre of a regular model, using the purity theorem in étale cohomology. As an illustration, we describe how the Brauer group of a smooth del Pezzo surface is determined by the singularity type of its reduction. We then relate the evaluation of an element of the Brauer group to the existence of points on certain torsors over the special fibre; we use this to describe situations when the evaluation is constant, and situations when the evaluation is surjective. In the latter case, we describe how thi…

A toroidal resolution for the bad reduction of some Shimura varieties

Borrowing a reduction principle to a recent preprint of G. Faltings (toroidal resolution of some matrix singularities, 1999), we use Lafforgue's compactification of PGL_r^{N+1}/PGL_r to construct a canonical log-smooth toroidal resolution for the bad reduction in a prime p of Shimura varieties of unitary and symplectic type with parahoric level structures at p. Using this result, non-canonical semi-stable resolutions over Z_p[p^{1/ν}] can be derived.

Atkin--Lehner 商模曲线在坏约化素数处的商

Atkin--Lehner Quotients of Modular Curves at Primes of Bad Reduction

本文构造了模曲线 $X_0(N)$ 的 Atkin--Lehner 商在坏约化素数处的半稳定模型，并给出特殊纤维的显式描述与亏格公式。

On the bad reduction of certain U(2, 1) Shimura varieties

Let $E$ be a quadratic imaginary field and let $p$ be a prime which is inert in $E.$ We study three types of Picard modular surfaces in positive characteristic $p$ and the morphisms between them. The first Picard surface, denoted $S$, parametrizes triples $(A,ϕ,ι)$ comprised of an abelian threefold $A$ with an action $ι$ of the ring of integers $\mathcal{O}_{E}$, and a principal polarization $ϕ$. The second surface, $S_{0}(p)$, parametrizes, in addition, a suitably restricted choice of a subgroup $H\subset A[p]$ of rank $p^{2}$. The third Picard surface, $\widetilde{S}$, parametrizes triples $…

Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field

We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. We study minimality and semi-stability, considering what conditions imply minimality (Theorem 4.4) and whether semi-stable models and presentations are minimal, proving results in the degree two case (Theorems 4.6, 4.7). We prove the singular reduction of a semi-stable pre…

On the modularity of three Calabi-Yau threefolds with bad reduction at 11

We investigate the modularity of three non-rigid Calabi-Yau threefolds with bad reduction at 11 which arise as fibre products of rational elliptic surfaces. For this purpose, we apply a method by Serre to compare two-dimensional 2-adic Galois representations with uneven trace. Hereby, we associate two of the threefolds (or, more precisely, a two-dimensional piece of their middle cohomology group) to newforms of weight 4 and level 22 and 55, respectively. This enables us to compute the zeta-functions of these varieties up to the Euler factors of the bad primes.

Reduction theorem for lattice cohomology

The lattice cohomology of a plumbed 3--manifold $M$ associated with a connected negative definite plumbing graph is an important tool in the study of topological properties of $M$, and in the comparison of the topological properties with analytic ones when $M$ is realized as complex analytic singularity link. By definition, its computation is based on the (Riemann--Roch) weights of the lattice points of $\Z^s$, where $s$ is the number of vertices of the plumbing graph. The present article reduces the rank of this lattice to the number of `bad' vertices of the graph. (Usually the geometry/topol…

Good reduction of the Brauer-Manin obstruction

For a smooth and projective variety over a number field with torsion free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the primes dividing the order of the transcendental Brauer group can turn up in the description of the Brauer-Manin set.

Reduction and lifting of special metacyclic covers

Special covers are metacyclic covers of the projective line, with Galois group of order pm, which have a specific type of bad reduction to characteristic p. Such covers arise in the study of the arithmetic of Galois covers of the projective line with three branch points. Our results provide a classification of all special covers in terms of certain lifting data in characteristic p.

Stable reduction of three point covers

This note gives a survey on some results related to the stable reduction of three point covers, which were the topic of my talk at the {\em Journées Arithmetiques} 2003 in Graz.

多项式方程组的坏约简素数

Primes of bad reduction for systems of polynomial equations

研究多项式方程组在模素数p下的Chow形式可约简条件，证明当p不整除特定非零整数时，Chow形式可保持结构。

One-point covers of elliptic curves and good reduction

Raynaud gave a criterion for a branched $G$-cover of curves defined over a mixed-characteristic discretely valued field $K$ with residue characteristic $p$ to have good reduction in the case of either a three-point cover of $\mathbb{P}^1$ or a one-point cover of an elliptic curve. Specifically, such a cover has potentially good reduction whenever $G$ has a Sylow $p$-subgroup of order $p$ and the absolute ramification index of $K$ is less than the number of conjugacy classes of order $p$ in $G$. In the case of an elliptic curve, we generalize this to the case in which $G$ has an arbitrarily lar…

Good reduction and cyclic covers

We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties, and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general set-up for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley-Weil theorem for stacks.

Good reduction of three-point Galois covers

Michel Raynaud gave a criterion for a three-point G-cover f : Y \rightarrow X = P^1, defined over a p-adic field K, to have good reduction. In particular, if the order of a p-Sylow subgroup of G is p, and the number of conjugacy classes of elements of order p is greater than the absolute ramification index e of K, then f has potentially good reduction. We give a different proof of this criterion, which extends to the case where G has an arbitrarily large cyclic p-Sylow subgroup, answering a question of Raynaud. We then use the criterion to give a family of examples of three-point covers with g…

Stable reduction of finite covers of curves

Let K be the function field of a connected regular scheme S of dimension 1, and let f : X -> Y be a finite cover of projective smooth and geometrically connected curves over K with g(X) greater or equal to 2. Suppose that f can be extended to a finite cover {\mathcal X} -> {\mathcal Y} of semi-stable models over S (it is known that this is always possible up to finite separable extension of K). We prove that there then exists a (unique) minimal such cover. This gives a canonical way to extend X -> Y to a finite cover of semi-stable models over S.

$p$ 次覆盖的半稳定约化

Semistable reduction of covers of degree $p$

本文提出一种计算局部域上配备 $p$ 次态射到射影直线的曲线半稳定约化的方法，涵盖非伽罗瓦覆盖，并特别实现了 $p=3$ 平面四次曲线在 SageMath 中的约化，基于 Berkovich 解析几何与微分函数。

完美oid覆盖上的线丛：良约化情形

Line bundles on perfectoid covers: case of good reduction

本文研究完美oid空间的Picard群与Picard函子，证明良约化极限覆盖的Picard函子可由特殊纤维表示，并借此解决若干公开问题，同时构造了$\mathbb G_m$的Hodge--Tate谱序列。

Cyclic p-groups and semi-stable reduction of curves in equal characteristic p>0

In this paper we study the semi-stable reduction of $p$ and $p^2$-cyclic covers of curves in equal characteristic $p>0$. The main tool we use is the classical Artin-Schreier-Witt theory for $p^n$-cyclic covers in characteristic $p$. Although the results of this paper concern only the cases of degree $p$ and $p^2$-cyclic cover we develop the techniques and the framework in which the general cyclic case can be studied.

An analog to Deuring's criterion for good reduction of elliptic curves

In this paper we study the reduction of $p$-cyclic covers of the $p$-adic line ramified at exactly four points. For $p=2$ these covers are elliptic curves and Deuring has given a criterion for when such a curve has good reduction. Here we consider the case of $p>2$ and completely determine the stable model of the cover. In particular we obtain a finite extension $R'$ of $R$ necessary for the stable reduction to be defined. No additional conditions are imposed on the geometry of the branch locus and thus this work can be viewed as a first step towards understanding the situation where branch po…

Reduction of the Hurwitz space of metacyclic covers

We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a hypergeometric differential equation. This generalizes the result of Deligne- Rapoport on the reduction of the modular curve X(p).

Reduction of covers and Hurwitz spaces

In this paper we study the reduction of Galois covers of curves, from characteristic 0 to characteristic p. The starting point is a is a recent result of Raynaud which gives a criterion for good reduction for covers of the projective line branch at 3 points. Under some condition on the Galois group, we extend this criterion to the case of 4 branch points. Moreover, we describe the reduction of the Hurwitz space of such covers and compute the number of covers with good reduction.

Stable reduction of modular curves

We determine the stable reduction at $p$ of all three point covers of the projective line with Galois group ${\rm SL}_2(p)$. As a special case, we recover the results of Deligne and Rapoport on the reduction of the modular curves $X_0(p)$ and $X_1(p)$. Our method does not use the fact that modular curves are moduli spaces. Instead, we rely on results of Raynaud and the authors which describe the stable reduction of three point covers whose Galois group is strictly divisible by $p$.

Another proof of the Semistable Reduction Theorem

We give a new proof of the Semistable Reduction Theorem for curves. The main idea is to present a curve $Y$ over a local field $K$ as a finite cover of the projective line $X=\PP^1_K$. By successive blowups (and after replacing $K$ by a suitable finite extension) we construct a semistable model of $X$ whose normalization with respect to the cover is a semistable model of $Y$.

广义马尔可夫曲面上的稳定约化与大尼尔森轨道

Stable reduction and large Nielsen orbits on generalized Markoff surfaces

该研究为广义马尔可夫曲面的$\text{SL}(2,\mathbb{Z})$轨道基数建立下界，结合相关结果确立分类猜想，扩展早期工作，通过非同余模曲线的稳定约化理论推导轨道基数下界。

Galois covers of degree p: semi-stable reduction and Galois action

In this paper we study the semi-stable reduction of Galois covers of degree p above semi-stable curves over a complete discrete valuation ring of inequal characteristics (0,p). We are also able to describe the Galois action on these covers in terms of some geometric and combinatorial datas in characteristic p endowed with the action of the Galois group of the residue field.

On the flatness of local models for the symplectic group

We investigate the bad reduction of certain Shimura varieties (associated to the symplectic group). More precisely, we look at a model of the Shimura variety at a prime p, with parahoric level structure at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced.