Dragos Ghioca
We determine the structure of the intersection of a finitely generated subgroup of a semiabelian variety $G$ defined over a finite field with a closed subvariety $X\subset G$.
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Using the Skolem-Mahler-Lech theorem, we prove a dynamical Mordell-Lang conjecture for semiabelian varieties.
The Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic
本文研究正特征代数闭域上半阿贝尔簇的 Mordell-Lang 问题，刻画其子簇与有理点有限生成子群的交集。
The isotrivial case in the Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic
本文研究了素特征代数闭域上定义的半阿贝尔簇在 Mordell-Lang 猜想中的同平凡情形，描述了其子簇与有限生成子群的交集。
The Mordell-Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell-Lang conjecture remains true in this setting if either (1) the semiabelian variety is simple, (2…
We here aim to complete our model-theoretic account of the function field Mordell-Lang conjecture, avoiding appeal to dichotomy theorems for Zariski geometries, where we now consider the general case of semiabelian varieties. The main result is a reduction, using model-theoretic tools, of the semiabelian case to the abelian case.
Uniform Mordell--Lang conjecture for semiabelian varieties
本文证明了半阿贝尔簇的一致Mordell-Lang猜想，为该猜想在更广泛簇类上的成立提供了完整证明。
We formulate a conjecture for semiabelian varieties A over number fields that includes both the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture. We prove the "Mordellic" (finitely generated) part of the conjecture when A is isogenous to the product of an abelian variety and a torus. The proof makes use of the Mordell-Lang conjecture, the Bogomolov conjecture, and an equidistribution theorem.
Let K be an algebraically closed field of prime characteristic p, let X be a semiabelian variety defined over a finite subfield of K, let f be a regular self-map on X defined over K, let V be a subvariety of X defined over K, and let x be a K-point of X. The Dynamical Mordell-Lang Conjecture in characteristic p predicts that the set S consisting of all positive integers n for which f^n(x) lies on V is a union of finitely many arithmetic progressions, along with finitely many p-sets, which are sets consisting of all integers of the form c_1 p^{k_1 n_1} + ... + c_m p^{k_m n_m} (as we vary n_1,..…
In "Mordell-Lang plus Bogomolov", we formulated a conjecture combining the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture, and proved the Mordellic part when A is almost split (i.e., isogenous to the product of an abelian variety and a torus). Here we prove that the full conjecture would follow from the Bogomolov conjecture and an equidistribution result for general semiabelian varieties. These assumptions have been proved when A is almost split, so the conjecture is completely proven in that case.
The Mordell-Lang conjecture (proven by Faltings, Vojta and McQuillan) states that the intersection of a subvariety $V$ of a semiabelian variety $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$ with a finite rank subgroup $Γ\le G(\mathbb{k})$ is a finite union of cosets of subgroups of $Γ$. We explore a variant of this conjecture when $G$ is a product of an abelian variety $A$ defined over $\mathbb{k}$ with the additive group $\mathbb{G}_a$.
Mordell--Weil groups over large algebraic extensions of fields of characteristic zero
本文研究了特征零域的大型代数扩张上半阿贝尔簇的Mordell--Weil群结构，发现新的Kummer-忠实域例子及其Mordell--Weil群模 torsion 的不同性质。
We prove a dynamical version of the Mordell-Lang conjecture for etale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer positively a question of Keeler/Rogalski/Stafford for critically dense sequences of closed points of a Noetherian integral scheme.
Kummer-faithful fields with finitely generated absolute Galois group
本文证明了在有限生成的绝对伽罗华群条件下，几乎所有数字段都是Kummer-忠实的，并展示了其在半阿贝尔簇Mordell-Weil群中的应用。
Hecke orbits and the Mordell-Lang conjecture in distinguished categories
本文在杰出范畴框架中引入$Σ$-轨道概念，统一了Shimura簇中的Hecke轨道与半阿贝尔簇中的有限秩子群，并借此证明Mordell-Lang型一般命题及Legendre族和$\mathcal{A}_g$中的无条件不太可能相交结果。
We prove a quantitative partial result in support of the Dynamical Mordell-Lang Conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field $K$ of characteristic $p$, given a semiabelian variety $X$ defined over a finite subfield of $K$ and endowed with a regular self-map $Φ:X \longrightarrow X$ defined over $K$, given a point $α\in X(K)$ and a subvariety $V\subseteq X$, then the set of all non-negative integers $n$ such that $Φ^n(α)\in V(K)$ is a union of finitely many arithmetic progressions along with a subset $S$ with the p…
Effective isotrivial Mordell-Lang in positive characteristic
本文利用有限自动机使正特征等平凡 Mordell-Lang 定理有效化并推广至一般交换代数群，进而给出点集增长二分定理及三类丢番图问题的判定程序。
We give a new proof of the Mordell-Lang conjecture in positive characteristic for finitely generated subgroups. We also make some progress towards the full Mordell-Lang conjecture in positive characteristic.
We prove a dynamical version of the Mordell-Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to the ones employed by Skolem, Chabauty, and Coleman for studying diophantine equations.
We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism f:X --> X. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics.
The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. M. McQuillan gave a proof of this statement. We revisit his proof, indicating some simplifications. This text contains a complete elementary proof of the fact that (GML) for groups of torsion points (= generalized Manin-Mumford conjecture), together with (GML) for finitely generated groups imply the full generalized Mordell-Lang conjecture.
Mordell--Lang plus Geometric Bogomolov
本文证明了特征零半阿贝尔簇上 Poonen 定理的几何版本，推广了 Mordell--Lang 与几何 Bogomolov 猜想。
Let $ϕ$ be a Drinfeld module of generic characteristic, and let $X$ be a sufficiently generic affine subvariety of $\mathbb{G}_a^g$. We show that the intersection of $X$ with a finite rank $ϕ$-submodule of $\mathbb{G}_a^g$ is finite.
A note on Bremner's conjecture and uniformity
本文通过高度统一的Mordell定理，作者给出了Bremner猜想中关于有理数域上椭圆曲线的有理点序列的长度与秩之间的关系的更直接证明，并扩展了这些方法到有限生成乘法群和几何等比数列的应用。
Uniform Mordell-Lang Plus Bogomolov
本文证明了阿贝尔簇上 Poonen“Mordell-Lang 加 Bogomolov”定理的一致版本，主要推广 Rémond 的大点结果以纳入额外 $ε$-邻域，小点部分基于与 Gao 和 Kühne 的早期合作。
In this paper we show how some known weak forms of the Zilber--Pink conjecture can be strengthened by combining them with the Mordell--Lang conjecture or its variants. We illustrate this idea by proving some theorems on atypical intersections in the semiabelian and modular settings. Given a "finitely generated" set $Γ$ with a certain structure, we consider $Γ$-special subvarieties -- weakly special subvarieties containing a point of $Γ$ -- and show that every variety $V$ contains only finitely many maximal $Γ$-atypical subvarieties, i.e. atypical intersections of $V$ with $Γ$-special varieties…
This expository survey is based on my online talk at the ICCM 2020. It aims to sketch key steps of the recent proof of the uniform Mordell-Lang conjecture for curves embedded into Jacobians (a question of Mazur). The full version of this conjecture is proved by combining Dimitrov-Gao-Habegger (https://annals.math.princeton.edu/articles/17715) and Kühne (arXiv:2101.10272). We include in this survey a detailed proof on how to combine these two results, which was implicitly done in another short paper of Dimitrov-Gao-Habegger (arXiv:2009.08505) but not explicitly written in existing literature. A…
Consider a smooth, geometrically irreducible, projective curve of genus $g \ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of $g$, $d$, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounded, in $g$ and $d$, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generaliz…
A Buium--Coleman bound for the Mordell--Lang conjecture
本文针对亏格 $g$ 双曲曲线，在好约化条件下给出 Mordell--Lang 轨迹的显式上界，并证明其非分歧性，进而推出 Mordell 蕴含 Mordell--Lang 猜想的新证明。
The Uniform Mordell-Lang Conjecture
本文证明了关于阿贝尔簇的统一莫尔德尔-兰猜想，通过推广曲线嵌入其雅可比簇中的间隙原理，得出了不依赖于阿贝尔簇的余数结果。
Dynamical Mordell-Lang problem for automorphisms of surfaces in positive characteristic
本文在正特征下研究了曲面自同构的动力学Mordell-Lang问题，通过分析动力系统与代数几何的相互作用，揭示了有限映射下的轨道结构与代数簇性质之间的联系。
Dynamical Mordell-Lang conjecture for split self-maps of affine curve times projective curve
本文证明了在仿射曲线与投影曲线乘积上，关于有理数闭包上内射映射的动力学Mordell-Lang猜想。
We note that Pillay's result on the stability of an algebraically closed field with a predicate for a group of Lang type implies that number uniformity follows formally from the finiteness results analogous to Faltings' Theorem.
We prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture, in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group. In particular, in the setting of the last sentence, we provide a proof of the Mordell-Lang conjecture, which does not depend on tools coming from model theory.
Adelic Mordell-Lang and the Brauer-Manin obstruction
本文研究整体函数域上阿贝尔簇闭子簇的有理点分布，证明满足条件的阿代尔点是子簇有理点的极限，结合Tate-Shafarevich群有限假设可得有理点在Brauer集中稠密，并将结果推广至全虚数数域情形。
We provide an explicit formula on the growth rate of ample heights of rational points under iteration of endomorphisms on smooth projective varieties over number fields. As an application, we give a positive answer to a problem of Dynamical Mordell-Lang type for pairs of étale endomorphisms, which is a variant of the original one stated by Bell, Ghioca, and Tucker in their monograph.
We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang conjecture.
The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group $Γ$ equipped with an arbitrary endomorphism F. This is applied to the isotrivial positive characteristic Mordell-Lang context where F is the Frobenius action on a commutative algebraic group G over a finite field, and $Γ$ is a finitely generated F-invariant subgroup of G. It is shown that the F-subsets of $Γ$ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X is a closed subva…
We present a reduction of the function field Mordell-Lang conjecture to the function field Manin-Mumford conjecture, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski structures. In this version 2, the quantifier elimination result in positive characteristic is extended from simple abelian varieties to all abelian varieties, completing the main theorem in the positive characteristic case. In version 3, some corrections are made to the proof of quantifier elimination in positive characteristic, and the paper is substantially reor…
We prove that two finite endomorphisms of the unit disk with degree at least two have orbits with infinite intersections if and only if they have a common iteration.
On Dynamical Mordell-Lang Conjecture for Split Map
本文证明了复数域上曲线自同态乘积的动力 Mordell-Lang 猜想，并给出了等差数列公差的某种可有效计算性。
We give an overview of some landmark theorems and recent conjectures in Diophantine Geometry. In the elliptic case, we prove some new bounds for torsion anomalous points and we clarify the implications of several height bounds on the effective Mordell-Lang Conjecture. We also explicitly bound the Neron-Tate height of the rational points of a family of curves of increasing genus.
We study the quasi-endomorphism ring of infinitely definable subgroups in separably closed fields. Based on the results we obtain, we are able to prove a Mordell-Lang theorem for Drinfeld modules of finite characteristic. Using specialization arguments we are able to prove also a Mordell-Lang theorem for Drinfeld modules of generic characteristic.
In previous work, the first author, Ghioca, and the third author introduced a broad dynamical framework giving rise to many classical sequences from number theory and algebraic combinatorics. Specifically, these are sequences of the form $f(Φ^n(x))$, where $Φ\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $\overline{\mathbb{Q}}$ and $x\in X(\overline{\mathbb{Q}})$ is a point whose forward orbit avoids the indeterminacy loci of $Φ$ and $f$. They conjectured that if the sequence is infinite, then $\limsup \frac{h(f(Φ^n(x)))}{\log n} > 0$. They also made a correspondi…
Around the dynamical Mordell-Lang conjecture
本文报告了动力学Mordell-Lang猜想的进展，将其已知结果推广至复数域及一致与部分轨道版本，并利用Berkovich空间语言改进了p-adic方法，同时提出了进一步问题。
The dynamical Mordell-Lang conjecture concerns the structure of the intersection of an orbit in an algebraic dynamical system and an algebraic variety. In this paper, we bound the size of this intersection for various cases when it is finite.
We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $ϕ$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $ϕ$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(\bP^1)^g$ has only finite intersection with any curve contained in $(\bP^1)^g$. We also show that our result holds for indecomposable polynomials $ϕ$ with coefficients in $\bC$. Our proof uses results from $p$-adic dynamics together with an integrality argume…
We offer some theorems, mainly of finiteness, for certain patterns in elliptical billiards, related to periodic trajectories. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in $(0,π)$, there are only finitely many cases for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results have their origin in `relative' cases of the Manin-Mumford conjecture, and constitute instances of how arithmetical content may affect…
Let K be an algebraically closed field of prime characteristic p, let N be a positive integer, let f be a self-map on the algebraic torus T=G_m^N defined over K, let V be a curve in T defined over K, and let x be a K-point of T. We show that the set S consisting of all positive integers n for which f^n(x) is contained in V is a union of finitely many arithmetic progressions, along with a finite set and with finitely many p-arithmetic sequences, which are sets of the form {b + ap^{kn}: n is a positive integer} where a and b are given rational numbers and k is a positive integer. We also prove t…
We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to a question of Mazur's concerning the uniformity of the Mordell-Lang conjecture for curves. In particular we show that the relative Bogomolov conjecture implies the uniform Manin-Mumford conjecture for curves. The proof is built up on our previous work "Uniformity in Mordell-Lang for curves".
Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point on X. We show that the set S consisting of all nonnegative integers n such that f^n(x) is in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given any quasi-projective variety X, any rational self-map map f on X, any subvariety Y of X, and any point x in X whose orbit under f is in the domain of definition for f, the set S is a finite union of arithmetic progressions together with a set of Bana…
We show that if $ϕ: X \to X$ is an automorphism of a smooth projective variety and $D \subset X$ is an irreducible divisor for which the set of $d$ in $D$ with $ϕ^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X, ϕ)$ admits an equivariant rational fibration to a curve. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of $\textrm{Aut}(X)$, extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety …
Let k be an algebraically closed field of characteristic 0, let X=P^1\times A^N and let f be a rational endomorphism of X given by (x,y)--->(g(x), A(x)y), where g is a rational function, while A is an N-by-N matrix with entries in k(x). We prove that if g is of the form x--->ax+b, then each irreducible curve C of X which intersects some orbit of f in infinitely many points must be periodic under the action of f. In the case g is an endomorphism of degree greater than 1, then we prove that each irreducible subvariety Y of X intersecting an orbit of f in a Zariski dense set of points must be per…
Dynamical GCD Problems and a Variant of the Dynamical Mordell-Lang Conjecture
本文研究了动力学GCD问题，提出了自同构情况下的新方法，并证明了与高维动力学Mordell-兰猜想相关的结果。
Height arguments toward the dynamical Mordell-Lang problem in arbitrary characteristic
本文利用高度论证方法，证明了动力学Mordell-Lang问题中两个关键结果：关于投影变种中稠密轨道返回集的有限性及产品结构下的返回集形式一致性，揭示了动力系统中返回集的复杂性。
Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type
利用Presburger算术理论，证明多项式对(f,g)的秩二递归集是半线性的，推广了Yang和Zhong关于m=n情形的定理，并得到有理映射在m=n情形下的部分结果。
The Dynamical Mordell--Lang Conjecture for Relatively Étale Systems over Products of Curves
本文证明了复数域上相对平展自态射的动力 Mordell--Lang 猜想，涵盖曲线乘积的任意自态射，通过算术输入与 p 进方法结合实现。
Let F : P^n --> P^n be a morphism of degree d > 1 defined over C. The dynamical Mordell--Lang conjecture says that the intersection of an orbit O_F(P) and a subvariety X of P^n is usually finite. We consider the number of linear subvarieties L in P^n such that the intersection of O_F(P)and L is "larger than expected." When F is the d'th-power map and the coordinates of P are multiplicatively independent, we prove that there are only finitely many linear subvarieties that are "super-spanned" by O_F(P), and further that the number of such subvarieties is bounded by a function of n, independent o…
Dynamical Mordell--Lang Conjecture for Higher-Rank Radially Ramified Skew Products
本文证明了复数域上径向分歧多项式斜积族的动力 Mordell--Lang 猜想，并建立了统一的 Skolem--Mahler--Lech 定理，适用于轨道与线性递推的联合多项式与有理关系。
Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $ϕ$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the only such functions are those of the form $cx^j(ψ(x))^m$ with $ψ\in K(x)$, and for $m \leq 4$ we show the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before. With additional analysis, we show that the index set $\{n \geq 0 : ϕ^{n}(a) \in λ(\mathbb{P}^1(K))\}$ is a…
A converse of dynamical Mordell--Lang conjecture in positive characteristic
该研究证明了正特征下动力Mordell--Lang猜想的逆命题，构造出对应的分裂环面、自同态、点与闭子簇，使其迭代轨道与闭子簇的交点下标集恰好为给定的由等差数列和p-集合构成的并集。
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