Approximation at places of bad reduction

p-adic integration on bad reduction hyperelliptic curves

In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser fo…

模曲线与不良约减

Modular curves and bad reduction

研究证明在不同假设下，所有模曲线上的点对应的椭圆曲线在特定素数处有不良约减，例如定义在有理数域上的椭圆曲线在特定素数处有不良约减。

Introduction to Shimura varieties with bad reduction of parahoric type

This survey article explains the construction of Rapoport-Zink local models and their use in understanding various questions relating to the singularities in the reduction modulo p of certain Shimura varieties with parahoric level structure at p.

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.

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…

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。

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…

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…

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 $…

Compactification de varietes de Siegel aux places de mauvaise reduction

We construct arithmetic toroidal compactifications of the moduli stack of principally polarized abelian varieties with parahoric level structure. To this end, we extend the methods of Faltings and Chai to a case of bad reduction. ----- Nous construisons des compactifications toroidales arithmétiques du champ de modules des variétés abéliennes principalement polarisées munies d'une structure de niveau parahorique. Pour ce faire, nous étendons la méthode de Faltings et Chai à un cas de mauvaise réduction.

Compactification minimale et mauvaise reduction

We construct the minimal compactification of some modular Siegel varieties at their bad reduction places. These varieties parametrize principally polarized abelian schemes endowed with a parahoric level structure at a prime number $p$, and with an auxiliary level structure ; such varieties have bad reduction at $p$. We also sketch the arithmetic theory of the associated Siegel modular forms. ----- Nous construisons la compactification minimale de certaines varietes modulaires de Siegel en leurs places de mauvaise reduction. Ces varietes parametrent des schemas abeliens principalement polarises…

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…

Height pairings on Shimura curves and p-adic uniformization

We establish a relation between intersection numbers of special cycles on a Shimura curve and special values of derivatives of metaplectic Eisenstein series at a place of bad reduction where p-adic uniformization in the sense of Cherednik and Drinfeld holds. The result extends the one established by one of us (S. Kudla: Ann. of Math. 146 (1997)) for the archimedean place and for the non-archimedean places of good reduction. The bulk of the paper is concerned with the corresponding problem on the Drinfeld upper half plane (the formal scheme version).

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。

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.

Weak approximation and rationally connected varieties over function fields of curves

This is a survey of weak approximation over complex function fields, touching on the Koll'ar-Miyaoka-Mori theorem, places of good and bad reduction, the special case of rational surfaces, rationally simply connected varieties, and applications to low-degree complete intersections.

Good Reduction of Good Filtrations at Places

We consider filtered or graded algebras $A$ over a field $K$. Assume that there is a discrete valuation $O_v$ of $K$ with $m_v$ its maximal ideal and $k_v:=O_v/m_v$ its residue field. Let $Λ$ be $O_v$-order such that $ΛK=A$ and $\barΛ:=k_v\otimes_{O_v}Λ$ the $Λ$-reduction of $A$ at the place $K\leadsto k_v$. Using the filtration of $A$ induced by $Λ$ we shall prove that for certain algebras $A$ their properties are related to $\barΛ$.

Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

We consider the reduction of Brauer classes on surfaces over number fields, with a view toward applications to rationality and derived equivalence. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial upon reduction for a set of places of positive natural density. As a consequence, there are cubic fourfolds which become rational upon reduction for a positive proportion of places, and there are twisted derived equivalent K3 surfaces which become derived equivalent upon reduction for a positive proportion of places.

Ordinary reduction of K3 surfaces

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension L/K such that X has ordinary reduction at every non-archimedean place of L outside a density zero set of places.

Good reduction of algebraic groups and flag varieties

In 1983, Faltings proved that there are only finitely many abelian varieties over a number field of fixed dimension and with good reduction outside a given set of places. In this paper, we consider the analogous problem for other algebraic groups and their homogeneous spaces, such as flag varieties.