Dragos Ghioca
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.
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In this paper we prove a special case of the Lehmer inequality for Drinfeld modules. Also, based on this inequality, we prove certain Mordell-Weil type of theorems for certain infinitely generated fields.
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.
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.
In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers ${\cal O}_K$ of $t$-modules that are products of the Drinfeld modules ${\widehatφ}=ϕ_{1}^{e_1}\times \dots \times ϕ_{t}^{e_{t}}.$ Here $K$ is a finite extension of the field of fractions of $A={\mathbb F}_{q}[t].$ We assume that the ${\mathrm{rank}}(ϕ)_{i})=d_{i}$ and endomorphism rings of the involved Drinfeld modules of generic characteristic are the simplest possible, i.e. ${\mathrm{End}}(ϕ_{i})=A$ for $ i=1,\dots , t.$ Our main result is the following numeric criterion. Let ${N}=…
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.
It is conjectured that for fixed $A$, $r \ge 1$, and $d \ge 1$, there is a uniform bound on the size of the torsion submodule of a Drinfeld $A$-module of rank $r$ over a degree $d$ extension $L$ of the fraction field $K$ of $A$. We verify the conjecture for $r=1$, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of $K$. Moreover, we show that within an $\Lbar$-isomorphism class, there are only finitely many Drinfeld modules up to isomorphism over $L$ which have nonzero torsion. For the case $A=\Fq[T]$, $r=1$, and $L=\Fq(T)$, we give …
If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is nonzero. This represents the analogue in the function field setting of a theorem for rational elliptic curves due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof e…
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.
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.
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 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 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 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 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.
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.
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.
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。
Using the Skolem-Mahler-Lech theorem, we prove a dynamical Mordell-Lang conjecture for semiabelian varieties.
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。
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.
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。
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 $X\subset G$.
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 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 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。
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。
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 $\mathbb{G}_a$.
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…
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…
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。
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…
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…
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。
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…
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。
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。
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…
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。
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…
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,..…
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。
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。
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…
This article is the last one about the isomorphism between Lubin-Tate and Drinfeld towers. We prove the existence of an isomorphism between the compactly supported etale cohomology of the Lubin-Tate and Drinfeld towers, and more generally their equivariant cohomology complex. We also prove the existence of a geometric local Jacquet-Langlands correspondence between some equivariant rigid etale sheaves on Gross-Hopkins period space $\mathbb{P}^{n-1}$ and Drinfeld one $Ω$.
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 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…
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…
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…
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。
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…
We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic, let $G$ be a finitely generated subgroup of the multiplicative group of $K$, and let $X$ be a (irreducible) quasiprojective variety defined over $K$. We consider $K$-valued sequences of the form $a_n:=f(φ^n(x_0))$, where $φ\colon X\rightarrow X$ and $f\colon X\rightarrow\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci…
We prove an analogue of the Sato-Tate conjecture for Drinfeld modules. Using ideas of Drinfeld, J.-K. Yu showed that Drinfeld modules satisfy some Sato-Tate law, but did not describe the actual law. More precisely, for a Drinfeld module ϕdefined over a field L, he constructs a continuous representation ρ_\infty : W_L \to D^* of the Weil group of L into a certain division algebra, which encodes the Sato-Tate law. When the Drinfeld module has generic characteristic and L is finitely generated, we shall describe the image of this representation up to commensurability. As an application, we give i…
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。
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