Canonical height functions on the affine plane associated with polynomial automorphisms

Height of Some Automorphisms of Local Fields

In this note, we determine which automorphism subgroups of $\mathrm{Aut}_{\mathbb{F}_q}(\mathbb{F}_q((x)))$ are corresponding to $\mathbb{Z}_p$-extensions or $\mathbb{Z}_p\times\mathbb{Z}_p$-extensions of characteristic $0$ fields.

Sur la dynamique p-adique arithmétique des automorphismes de l'espace affine

We show that the set of periods of an automorphism of the affine plane defined over a $p$-adic field is bounded above by a constant independent from the automorphism. We deduce from this result a new proof in arithmetic dynamics of the Silverman's conjecture in dimension 2.

Automorphisms groups for $p$-cyclic covers of the affine line

Let $k$ be an algebraically closed field of positive characteristic $p>0$ and $C \to {\mathbb P}^1_k$ a $p$-cyclic cover of the projective line ramified in exactly one point. We are interested in the $p$-part of the full automorphism group $Aut_k C$. First we prove that these groups are exactly the extra-special $p$-groups and groups G which are subgroups of an extra-special group E such that $Z(E) \subseteq G$. The paper also describes an efficient algorithm to compute the $p$-part of $\Aut_k C$ starting from an Artin-Schreier equation for the cover $C \to {\mathbb P}^1_k$. The interest for t…

仿射曲面的斜驶自同构的轨道的交

Intersection of orbits of loxodromic automorphisms of affine surfaces

该研究证明域上仿射曲面的两个斜驶自同构若轨道有无穷多交点则必有公共迭代，借助前期仿射曲面自同态动力学工作与算术动力学论证，还得到乘积空间中曲面的动力学Mordell-Lang型结果。

仿射曲面双曲型自同构的周期点刚性

Rigidity of periodic points for loxodromic automorphisms of affine surfaces

本文证明仿射曲面上动力学次数大于1的自同构若共享稠密周期点集则周期点完全相同，通过构造典范高度并利用算术等分布理论（含Moriwaki高度情形）完成证明。

Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint

Let X be a smooth projective surface over a number field, and $f: X \to X$ an automorphism of positive topological entropy. In this paper, we construct a height function on X that behaves well relative to f and deduce some arithmetic properties of (X; f).

On the distribution of points of bounded height on equivariant compactifications of vector groups

We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space. (Nous établissons un développement asymptotique du nombre de points rationnels de hauteur bornée sur les compactifications équivariantes lisses de l'espace affine.)

仿射空间中模形的循环周期点及具有有界house和高度的周期点

Cyclotomic preperiodic points for morphisms in affine spaces and preperiodic points with bounded house and height

本文研究了仿射空间中模形的周期点集在循环闭包上的稠密性问题，并探讨了具有有界house和高度的周期点的性质。

通过 Mordell-Weil 格对称性获得的有效高度界

Effective height bounds via Mordell Weil lattice symmetries

本文利用 Jacobian 的 Mordell-Weil 格自同构与平均谱隙准则，建立数域上亏格至少二曲线有理点 Neron-Tate 高度的显式可计算上界，并给出秩 2 的 Bravais 格检验及算例。

The Geometric Dynamical Northcott Property For Regular Polynomial Automorphisms of the Affine Plane

We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $\mathbf{K}$ of characteristic zero, improving results of Ingram. For that, we show that when $\mathbf{K}$ is the field of rational functions of a smooth complex projective curve, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a si…

Points of bounded height on equivariant compactifications of vector groups, I

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.

具有无限离散自同构群的三次曲面

Cubic surfaces with infinite, discrete automorphism group

本文证明了方程为$xyz=g(x,y)$的仿射三次曲面的自同构群包含${\mathbb Z}$作为有限指数子群，此类方程最早由Mordell研究。

仿射平面上可交换多项式映射的自同构群

Automorphism Groups of Commuting Polynomial Maps of the Affine Plane

本文研究与秩为N的半单李代数相关联的多项式折叠映射，推导出A2、B2、G2对应的仿射平面上映射首项公式，并据此计算各折叠映射的仿射自同构群。

Height four formal groups with quadratic complex multiplication

We construct spectral sequences for computing the cohomology of automorphism groups of formal groups with complex multiplication by a $p$-adic number ring. We then compute the cohomology of the group of automorphisms of a height four formal group law which commute with complex multiplication by the ring of integers in the field $\mathbb{Q}_p(\sqrt{p})$, for primes $p>5$. This is a large subgroup of the height four strict Morava stabilizer group. The group cohomology of this group of automorphisms turns out to have cohomological dimension $8$ and total rank $80$. We then run the $K(4)$-local $E…

Rigidity of measurable structure for Z^d-actions by automorphisms of a torus

We show that for certain classes of actions of Z^d, d >= 2, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we consruct various examples of Z^d-actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomor…