Good Reduction of Periodic Points

Preperiodic points for rational functions defined over a global field in terms of good reductions

Let $ϕ$ be an endomorphism of the projective line defined over a global field $K$. We prove a bound for the cardinality of the set of $K$-rational preperiodic points for $ϕ$ in terms of the number of places of bad reduction. The result is completely new in the function fields case and it is an improvement of the number fields case. An important tool is an $S$-unit equation theorem in 2 variables.

Criteria for good reduction of proper polycurves

We give good reduction criteria for hyperbolic polycurves, i.e., successive extensions of families of curves, under mild assumption. These criteria are higher dimensional version of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa.

On a good reduction criterion for proper polycurves with sections

We give a good reduction criterion for proper polycurves with sections,i.e., successive extensions of family of curves with section, under mild assumption. This criterion is a higher dimensional version of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa.

Non-existence of elliptic curves with everywhere good reduction over some real quadratic fields

We prove the non-existence of elliptic curves having good reduction everywhere over some real quadratic fields.

A Prym variety with everywhere good reduction over $\mathbb{Q}(\sqrt{61})$

We compute an equation for a modular abelian surface $A$ that has everywhere good reduction over the quadratic field $K = \mathbb{Q}(\sqrt{61})$ and that does not admit a principal polarization over $K$.

Abelian surfaces with supersingular good reduction and non semisimple Tate module

We show the existence of abelian surfaces $A$ over $\mathbb{Q}_p$ having good reduction with supersingular special fibre whose associated $p$-adic Galois module $V_p(A)$ is not semisimple.

A criterion for good reduction of Drinfeld modules and Anderson motives in terms of local shtukas

For an Anderson A-motive over a discretely valued field whose residue field has A-characteristic ε, we prove a criterion for good reduction in terms of its associated local shtuka at ε. This yields a criterion for good reduction of Drinfeld modules. Our criterion is the function-field analog of Grothendieck's and de Jong's criterion for good reduction of an abelian variety over a discretely valued field with residue characteristic p in terms of its associated p-divisible group.

Elliptic Curves with Everywhere Good Reduction

We consider the question of which quadratic fields have elliptic curves with everywhere good reduction. By revisiting work of Setzer, we expand on congruence conditions that determine the real and imaginary quadratic fields with elliptic curves of everywhere good reduction and rational $j$-invariant. Using this, we determine the density of such real and imaginary quadratic fields. If $R(X)$ (respectively $I(X)$) denotes the number of real (respectively imaginary) quadratic fields $K=\mathbb{Q}[\sqrt{m}]$ such that $|Δ_K| < X$ and for which there exists an elliptic curve $E/K$ with rational $j$…

Elliptic curves with good reduction away from 2: III

We list the elliptic curves defined over $Q(\sqrt 5)$ with good reduction away from 2. There are 368 isomorphism classes. We give a global minimal model for each class.

The crystalline comparison of Ainf-cohomology: the case of good reduction

We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.

On some notions of good reduction for endomorphisms of the projective line

Let $Φ$ be an endomorphism of $\SR(\bar{\Q})$, the projective line over the algebraic closure of $\Q$, of degree $\geq2$ defined over a number field $K$. Let $v$ be a non-archimedean valuation of $K$. We say that $Φ$ has critically good reduction at $v$ if any pair of distinct ramification points of $Φ$ do not collide under reduction modulo $v$ and the same holds for any pair of branch points. We say that $Φ$ has simple good reduction at $v$ if the map $Φ_v$, the reduction of $Φ$ modulo $v$, has the same degree of $Φ$. We prove that if $Φ$ has critically good reduction at $v$ and the reduction…

二次域上具有处处好约化的亏格2曲线

Genus two curves with everywhere good reduction over quadratic fields

本研究针对二次域上二维绝对单阿贝尔簇的存在性问题，构造了首个由实/复二次域与对应域上处处好约化亏格2曲线组成的无穷序列，并证明其雅可比簇为绝对单阿贝尔簇。

Bound for preperiodic points for maps with good reduction

Let $K$ be a number field and let $ϕ$ in $K(z)$ be a rational function of degree $d\geq 2$. Let $S$ be the places of bad reduction for $ϕ$ (including the archimedan places). Let $Per(ϕ,K)$, $PrePer(ϕ, K)$, and $Tail(ϕ,K)$ be the set of $K$-rational periodic, preperiodic, and purely preperiodic points of $ϕ$, respectively. The present paper presents two main results. The first result gives a bound for $|PrePer(ϕ,K)|$ in terms of the number of places of bad reduction $|S|$ and the degree $d$ of the rational function $ϕ$. This bound significantly improves a previous bound given by J. Canci and L.…

非超奇异情形下广义Kummer曲面的好约化

The good reduction of generalized Kummer surfaces in the non-supersingular case

本文针对非超奇异情形，建立了阿贝尔曲面带循环群作用时广义Kummer曲面好约化的判别准则，推广了前人关于Kummer曲面的相关结果。

A Bogomolov property for the canonical height of maps with superattracting periodic points

We prove that if $f$ is a polynomial over a number field $K$ with a finite superattracting periodic point and a non-archimedean place of bad reduction, then there is an $ε>0$ such that only finitely many $P\in K^{\text{ab}}$ have canonical height less than $ε$ with respect to $f$. The key ingredient is the geometry of the filled Julia set at a place of bad reduction. We also prove a conditional uniform boundedness result for the $K$-rational preperiodic points of such polynomials, as well as a uniform lower bound on the canonical height of non-preperiodic points in $K$. We further prove uncond…

模曲线和志村曲线商中具有良好约化的意外素数

Unexpected primes of good reduction in quotients of modular and Shimura curves

研究权重2、平方自由层数N且有固定阿特金 - 勒纳符号模式的新形式零维空间，借此对平方自由层数的模曲线和志村曲线阿特金 - 勒纳商中良好约化的意外素数进行分类。

Examples of abelian surfaces with everywhere good reduction

We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler-Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures.

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.

多项式约简中周期点的比例

Proportion of periodic points in reduction of polynomials

研究多项式约简中周期点比例的极限行为，分析非线性共轭到切比雪夫多项式的多项式情况，得出极限为零的结论。

An example of a Brauer-Manin obstruction to weak approximation at a prime with good reduction

Following Bright and Newton, we construct an explicit K$3$ surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.

On the existence of abelian surfaces with everywhere good reduction

Let $D \le 2000$ be a positive discriminant such that $F = \mathbf{Q}(\sqrt{D})$ has narrow class one, and $A/F$ an abelian surface of ${\rm GL}_2$-type with everywhere good reduction. Assuming that $A$ is modular, we show that $A$ is either an $F$-surface or is a base change from $\mathbf{Q}$ of an abelian surface $B$ such that ${\rm End}_{\mathbf{Q}}(B) = \mathbf{Z}$, except for $D = 353, 421, 1321, 1597$ and $1997$. In the latter case, we show that there are indeed abelian surfaces with everywhere good reduction over $F$ for $D = 353, 421$ and $1597$, which are non-isogenous to their Galois…

Berkovich固定位的连通分量：潜在良好约简

Connected components of Berkovich fixed locus: Potential good reduction

本文研究了具有潜在良好约简性质的有理映射的固定位的拓扑结构，给出了其连通性和有限性的判据，并提供了固定位连通分量数的上界。

Generalized Kuga-Satake theory and good reduction properties of Galois representations

In previous work we described when a single geometric representation, valued in a linear algebraic group, of the Galois group of a number field lifts through a central torus quotient to a geometric representation. In this paper we prove a much sharper result for systems of l-adic representations, such as the l-adic realizations of a motive, having common "good reduction" properties. Namely, such systems admit geometric lifts with good reduction outside a common finite set of primes. The method yields new proofs of theorems of Tate (the original result on lifting projective representations over…

The supersingular locus of unitary Shimura varieties with exotic good reduction

In this paper, we use a group-theoretic approach to give a concrete description of the geometric structure of the supersingular locus of unitary Shimura varieties with exotic good reduction. This approach also is a more uniform way to prove results of this form obtained previously by, for example, Vollaard-Wedhorn and Rapoport-Terstiege-Wilson.

Potentially good reduction loci of Shimura varieties

In this paper, we give a notion of the potentially good reduction locus of a Shimura variety. It consists of the points which should be related with motives having potentially good reductions in some sense. We show the existence of such locus for a Shimura variety of preabelian type. Further, we construct a partition of the adic space associated to a Shimura variety of preabelian type, which is expected to describe degenerations of motives. Using this partition, we prove that the cohomology of the potentially good reduction locus is isomorphic to the cohomology of a Shimura variety up to non-s…

Good reduction criterion for K3 surfaces

We prove a Neron--Ogg--Shafarevich type criterion for good reduction of K3 surfaces, which states that a K3 surface over a complete discrete valuation field has potential good reduction if its $l$-adic cohomology group is unramified. We also prove a $p$-adic version of the criterion. (These are analogues of the criteria for good reduction of abelian varieties.) The model of the surface will be in general not a scheme but an algebraic space. As a corollary of the criterion we obtain the surjectivity of the period map of K3 surfaces in positive characteristic.

Picard curves over Q with good reduction away from 3

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined, and an application to a question of Ihara is discussed.

Good Reduction of K3 Surfaces

Let $K$ be the field of fractions of a local Henselian DVR with perfect residue field. Assuming potential semi-stable reduction, we show that an unramified Galois-action on second $\ell$-adic cohomology of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga-Satake Abelian varieties. On our way, we settle existence and termination of certain semi-stable flops in mixed characteristic, and study gr…

Good and semi-stable reductions of Shimura varieties

We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.

Potential good reduction of hyperelliptic curves

Let $K$ be a number field, and $g \geq 2$ a positive integer. We define $c_K(g)$ as the smallest integer $n$ such that there exist infinitely many $\overline{K}$-isomorphism classes of genus $g$ hyperelliptic curves $C/K$ with all Weierstrass points in $K$ having potentially good reduction outside $n$ primes in $K$. We show that $c_K(g) > π_{K, \textrm{odd}}(2g) + 1$, where $π_{K, \textrm{odd}}(n)$ denotes the number of odd primes in $K$ with norm no greater than $n$, as well as present a summary of various conditional and unconditional results on upper bounds for $c_K(g)$.

Reduction of polynomial dynamical systems modulo primes

We study the algebraic dynamical systems generated by triangular systems of rational functions and estimate the height growth of iterations generated by such systems. Further, using a result on the reduction modulo primes of systems of multivariate polynomials over the integers, we study the periodic points and the intersection of orbits of such dynamical systems over finite fields. 1

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…

关于具有良好普通简化的椭圆曲线的反圆环Selmer群的拟环理想

On the Fitting ideals of anticyclotomic Selmer groups of elliptic curves with good ordinary reduction

本文通过theta元素确定了反圆环$\mathbb{Z}_p$-扩展中Selmer群的拟环理想，证明了Kurihara主猜想的反圆环版本。

Elliptic curves with good reduction outside of the first six primes

We present a database of rational elliptic curves, up to Q-isomorphism, with good reduction outside {2,3,5,7,11,13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to curves in the set. We also discuss the connection to S-unit equations and the existence of rational elliptic curves with maximal conductor.

Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures

Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:\mathbb{P}^1_K\to\mathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $\mathbb{P}^1_{R_S}\to\mathbb{P}^1_{R_S}$. A finite Galois invariant subset $X\subset\mathbb{P}^1_K(\bar{K})$ has good reduction outside $S$ if its closure in $\mathbb{P}^1_{R_S}$ is étale over $R_S$. We study triples $(f,Y,X)$ with $X=Y\cup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many $\text{PGL}_2(R_S)$-equivalence cl…

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…

具有潜在良好约化的后临界有限多项式的无限野分歧树状表示

Infinitely wildly ramified arboreal representations for postcritically finite polynomials with potential good reduction

本文证明了对于数域上后临界有界且具有潜在良好约化的多项式，其树状表示在满足整除次数条件的有限位上方要么有限，要么无限野分歧。

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…

CM点处处具有良好约化

CM points have everywhere good reduction

本文证明每个Shimura簇都存在一个整模型，使得所有CM点都具有良好约化，并给出一个更强的局部刻画：具有潜在良好约化的点恰为可延拓为整点的点。

分圆扩张中具有好约化的$\mathbb{Q}_p$上椭圆曲线的挠点

Torsion of elliptic curves over $\mathbb{Q}_p$ with good reduction in cyclotomic extensions

本文研究分圆扩张下$\mathbb{Q}_p$上带好约化的椭圆曲线挠点结构问题，对任意素数p及对应分圆扩张情形，完成了相关有理点群挠子群的结构分类。

Algebraic groups with good reduction and the genus problem

We first provide an overview of several results dealing with the genus of a division algebra and highlight the role of ramification in its analysis. We then give a survey of recent developments on the genus problem for simple algebraic groups and its connections to the analysis of groups with good reduction.

计算亏格2曲线在几乎良好约化的奇素数处的欧拉因子

Computing Euler factors of genus 2 curves at odd primes of almost good reduction

针对Q上亏格2曲线C在几乎良好约化奇素数p处的欧拉因子计算问题，提出基于簇图理论的高效算法，避免正则模型计算，可处理Magma等系统现有算法无法解决的示例。

On the algebraic functional equation of the eigenspaces of mixed signed Selmer groups of elliptic curves with good reduction at primes above $p$

Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field which has good reduction at every prime above $p$. Under suitable assumptions, we prove that the $η$-eigenspace and the $\barη$-eigenspace of mixed signed Selmer group of the elliptic curve are pseudo-isomorphic. As a corollary, we show that the $η$-eigenspace is trivial if and only if the $\barη$-eigenspace is trivial. Our results can be thought as a reflection principle which relate an Iwasawa module in a given eigenspace with another Iwasawa module in a "reflected" eigenspace.

On good reduction of some K3 surfaces related to abelian surfaces

The Neron--Ogg--Safarevic criterion for abelian varieties tells that whether an abelian variety has good reduction or not can be determined from the Galois action on its l-adic etale cohomology. We prove an analogue of this criterion for some special kind of K3 surfaces (those which admit Shioda--Inose structures of product type), which are deeply related to abelian surfaces. We also prove a p-adic analogue. This paper includes Ito's unpublished result for Kummer surfaces.

Abelian varieties over Q with bad reduction in one prime only

Let l be a prime. We show that there do not exist any non-zero semi-stable abelian varieties over Q with good reduction outside l if and only if l=2, 3, 5, 7 or 13. We show that any semi-stable abelian variety over Q with good reduction outside l=11 is isogenous to a power of the Jacobian variety of the modular curve X_0(11). In addition we show that there do not exist any non-zero abelian varieties over Q with good reduction outside l acquiring semi-stable reduction at l over a tamely ramified extension if and only if l \le 5.

Good Reductions of Shimura Varieties of Hodge Type in Arbitrary Unramified Mixed Characteristic, Part II

We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type.

Potentially Good Reduction of Barsotti-Tate Groups

Let R be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K' of K. We prove that there exists a constant c which depends on the absolute ramification index e(K'/Q_p) and the height of G such that G has good reduction over K if and only if G[p^c] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's p-adic Ner…

源自具有潜在好约化的 $\mathbb{Q}_3$ 上椭圆曲线的 3-adic 表示

The 3-adic representations arising from elliptic curves over $\mathbb{Q}_3$ with potential good reduction

本文完整分类了同构于 $\mathbb{Q}_3$ 上椭圆曲线 Tate 模的潜在结晶 3-adic 表示，通过过滤 $(φ, \mathrm{Gal}(K/\mathbb{Q}_3))$-模描述，并重点关注野性潜在好约化的情况。

Galois representations, Mumford-Tate groups and good reduction of abelian varieties

Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent Q-rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without non…

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.

Complete intersections: Moduli, Torelli, and good reduction

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.

Kummer面在非超奇异情况下的模2约简

Reduction of Kummer surfaces modulo 2 in the non-supersingular case

研究Kummer面在非超奇异情况下模2的良约简条件，构造了代数空间模型和方案模型的等价性。

Recent developments in the theory of linear algebraic groups: Good reduction and finiteness properties

This is a survey article on some recent developments in the arithmetic theory of linear algebraic groups over higher-dimensional fields, written for the Notices of the AMS.

Logarithmic good reduction of abelian varieties

Let $K$ be a field which is complete for a discrete valuation. We prove a logarithmic version of the Néron-Ogg-Shafarevich criterion: if $A$ is an abelian variety over $K$ which is cohomologically tame, then $A$ has good reduction in the logarithmic setting, i.e. there exists a projective, log smooth model of $A$ over $\mathcal{O}_K$. This implies in particular the existence of a projective, regular model of $A$, generalizing a result of Künnemann. The proof combines a deep theorem of Gabber with the theory of degenerations of abelian varieties developed by Mumford, Faltings-Chai et al.