Jorge Pineiro
In the present work we establish the equality of the canonical metric of two commuting maps on an algebraic variety X. As a consequence the canonical height and measure associated to both maps are identical.
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In this paper we adapt some techniques developed for K3 surfaces, to study the geometry of a family of projective varieties in $\Pl_K^2 \times \Pl_K^2 \times \Pl_K^2$ defined as the intersection of a form of degree $(2,2,2)$ and a form of degree $(1,1,1)$. Members of the family will be equipped with dominant rational self-maps and we will study the actions of those maps on divisors and compute the first dynamical degrees of the composition of any pair.
The classical (m,k)-Landen transform F_{m,k} is a self-map of the field of rational functions C(z) obtained by forming a weighted average of a rational function over twists by m'th roots of unity. Identifying the set of rational maps of degree d with an affine open subset of P^{2d+1}, we prove that F_{m,0} induces a dominant rational self-map R_{d,m,0} of P^{2d+1} of algebraic degree m, and for 0 < k < m, the transform F_{m,k} induces a dominant rational self-map R_{d,m,k} of algebraic degree m of a certain hyperplane in P^{2d+1}. We show in all cases that R_{d,m,k} extends nicely to a map of …
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。
We study the dynamics of maps arising from the composition of two non-commuting involution on a K3 surface. These maps are a particular example of reversible maps, i.e., maps with a time reversing symmetry. The combinatorics of the cycle distribution of two non-commuting involutions on a finite phase space was studied by Roberts and Vivaldi. We show that the dynamical systems of these K3 surfaces satisfy the hypotheses of their results, providing a description of the cycle distribution of the rational points over finite fields. Furthermore, we extend the involutions to include the case where t…
We prove, for the canonical height defined by Silverman [15] on monomial maps, the existence of effective lower bounds for heights of points with Zariski dense orbit, for cases with endomorphisms induced by matrices with real Jordan form.
The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P \subset (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups.
Explicit Kodaira-Spencer maps over PEL Shimura varieties
本文通过Kodaira-Spencer映射构造两个canonical线丛间的映射，并显式计算其对线丛canonical度量的影响，为比较两个canonical线丛及其算术交数提供了具体方法。
We show that the canonical height function defined by Silverman does not have the Northcott finiteness property in general. We develop a new canonical height function for monomial maps. In certain cases, this new canonical height function has nice properties.
Let $d\ge 2$ be an integer, let $c(t)$ be any rational map, and let $f_t(z) := (z^d+t)/z$ be a family of rational maps indexed by t. For each algebraic number $t$, we let $h_{f_t}(c(t))$ be the canonical height of $c(t)$ with respect to the rational map $f_t$. We prove that the map $H(t):=h_{f_t}(c(t))$ (as $t$ varies among the algebraic numbers) is a Weil height.
On the Canonical Height Gap for Polynomial Maps and Portraits of Preperiodic Points of Polynomials with Height $0$
研究\(\mathbb{P}^{1}\)上多项式映射的典范高度与朴素高度差，建立与次数\(d\)无关的精确界，以此确定\(\mathbb{Q}\)上高度为0的多项式映射的有理预周期点集并分类其画像。
We study canonical heights for plane polynomial mappings of small topological degree. In particular, we prove that for points of canonical height zero, the arithmetic degree is bounded by the topological degree and hence strictly smaller than the first dynamical degree. The proof uses the existence, proved by Favre and the first author, of certain compactifications of the plane adapted to the dynamics.
We consider the arithmetic of Henon maps f(x, y)=(ay, x+f(y)) defined over number fields and function fields, usually with the restriction that a=1. We prove a result on the variation of Kawaguchi's canonical height in families of Henon maps, and derive from this a specialization theorem, showing that the set of parameters above which a given non-periodic point becomes periodic is a set of bounded height. Proving this involves showing that the only points of canonical height zero for a Henon map over a function field are those which are periodic (in the non-isotrivial case). In the case of qua…
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…
Let F : P^N --> P^N be a dominant rational map. The dynamical degree of F is the quantity d_F = lim (deg F^n)^(1/n). When F is defined over a number field, we define the arithmetic degree of an algebraic point P to be a_F(P) = limsup h(F^n(P))^(1/n) and the canonical height of P to be h_F(P) = limsup h(F^n(P))/n^k d_F^n for an appropriately chosen integer k = k_F. In this article we prove some elementary relations and make some deep conjectures relating d_F, a_F(P), and h_F(P). We prove our conjectures for monomial maps.
Given two rational maps $φ$ and $ψ$ on $\PP^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<φ,ψ>$ which is closely related to the canonical height functions $h_φ$ and $h_ψ$ associated to these maps. Our main results show a strong connection between the value of $<φ,ψ>$ and the canonical heights of points which are small with respect to at least one of the two maps $φ$ and $ψ$. Several necessary and sufficient conditions are given for the vanishing of $<φ,ψ>$. We give an explicit upper bound on the difference between the canonical height $h_ψ$ and the standard…
We consider a multi-parameter family of canonical coordinates and mirror maps o\ riginally introduced by Zudilin [Math. Notes 71 (2002), 604-616]. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated example of Candelas, de la Ossa, Green and\ Parkes [Nucl. Phys. B359 (1991), 21-74] associated to the quintic hypersurface in $\mathbb P^4(\mathbb C)$. In [Duke Math. J. 151 (2010), 175-218], we proved that all coeffi\ cients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the correspo…
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