Mordell-Lang Conjecture Research Articles

Periodic points and arithmetic degrees of certain rational self-maps

Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the second dynamical degrees distinct. We give a boundedness result about heights of its periodic points. This is motivated by a conjecture of Silverman for polynomial automorphisms of affine spaces. We also study the Kawaguchi–Silverman conjecture concerning dynamical and arithmetic degrees for certain rational self-maps in dimension two. In particular, we reduce the problem to the dynamical Mordell–Lang conjecture and verify the Kawaguchi–Silverman conjecture for some new cases. As a byproduct of the argument, we show the existence of Zariski dense orbits in these cases.

Backward orbits of critical points

We examine the Galois groups of the extensions K((f′∘fn)−1(0))/K where K is a number field for polynomials f(x)∈K[x]. We use our understanding of this group to study the proportion of primes for which f has a p-adic attracting periodic point for a “typical” f and apply the result to the split case of the Dynamical Mordell-Lang Conjecture.

Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture

AbstractWe offer some theorems, mainly finiteness results, for certain patterns in elliptical billiards, related to periodic trajectories; these seem to be the first finiteness results in this context. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in , there are only finitely many directions 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 (which are shown not to hold for general billiards) have their origin in ‘relative’ cases of the Manin–Mumford conjecture and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell–Lang conjecture. In turn, this variant embraces cases, which, somewhat surprisingly, sometimes can be treated (only) by completely different methods compared to the former ones; here we shall offer an explicit example related to diophantine equations in algebraic tori.

Perfect points of abelian varieties

Let $p$ be a prime number, $k$ a finite field of characteristic $p>0$ and $K/k$ a finitely generated extension of fields. Let $A$ be a $K$ -abelian variety such that all the isogeny factors are neither isotrivial nor of $p$ -rank zero. We give a necessary and sufficient condition for the finite generation of $A(K^{\mathrm {perf}})$ in terms of the action of $\mathrm {End}(A)\otimes \mathbb {Q}_p$ on the $p$ -divisible group $A[p^{\infty }]$ of $A$ . In particular, we prove that if $\mathrm {End}(A)\otimes \mathbb {Q}_p$ is a division algebra, then $A(K^{\mathrm {perf}})$ is finitely generated. This implies the ‘full’ Mordell–Lang conjecture for these abelian varieties. In addition, we prove that all the infinitely $p$ -divisible elements in $A(K^{\mathrm {perf}})$ are torsion. These reprove and extend previous results to the non-ordinary case.

𝑝-adic interpolation of orbits under rational maps

Let L L be a field of characteristic zero, let h : P 1 → P 1 h:\mathbb {P}^1\to \mathbb {P}^1 be a rational map defined over L L , and let c ∈ P 1 ( L ) c\in \mathbb {P}^1(L) . We show that there exists a finitely generated subfield K K of L L over which both c c and h h are defined along with an infinite set of inequivalent non-archimedean completions K p K_{\mathfrak {p}} for which there exists a positive integer a = a ( p ) a=a(\mathfrak {p}) with the property that for i ∈ { 0 , … , a − 1 } i\in \{0,\ldots ,a-1\} there exists a power series g i ( t ) ∈ K p [ [ t ] ] g_i(t)\in K_{\mathfrak {p}}[[t]] that converges on the closed unit disc of K p K_{\mathfrak {p}} such that h a n + i ( c ) = g i ( n ) h^{an+i}(c)=g_i(n) for all sufficiently large n n . As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps ( h , g ) (h,g) of P 1 × X \mathbb {P}^1 \times X with g g an étale self-map of a quasiprojective variety X X .

Dynamical Mordell–Lang conjecture for totally inseparable liftings of Frobenius

Dynamical Mordell–Lang conjecture for totally inseparable liftings of Frobenius

Unlikely intersections on the p-adic formal ball

We investigate generalizations along the lines of the Mordell–Lang conjecture of the author’s p-adic formal Manin–Mumford results for n-dimensional p-divisible formal groups mathcal {F}. In particular, given a finitely generated subgroup Gamma of mathcal {F}({overline{mathbb {Q}}}_p) and a closed subscheme Xhookrightarrow mathcal {F}, we show under suitable assumptions that for any points Pin X(mathbb {C}_p) satisfying nPin Gamma for some nin mathbb {N}, the minimal such orders n are uniformly bounded whenever X does not contain a formal subgroup translate of positive dimension. In contrast, we then provide counter-examples to a full p-adic formal Mordell–Lang result. Finally, we outline some consequences for the study of the Zariski-density of sets of automorphic objects in p-adic deformations. Specifically, we do so in the context of the nearly ordinary p-adic families of cuspidal cohomological automorphic forms for the general linear group constructed by Hida.

Unlikely intersection problems for restricted lifts of a p-th power

We applied the theory of perfectoid spaces to prove dynamical versions of the Manin-Mumford conjecture, Mordell-Lang conjecture, and Tate-Voloch conjecture for lifts of a p-th power, generalizing the work of Xie in [39] for lifts of Frobenius.

Remarks on algebraic dynamics in positive characteristic

AbstractIn this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. Applying the arithmetic degree and canonical height in positive characteristic, we prove the Dynamical Mordell–Lang Conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit Conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell–Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.

A Galois–Dynamics Correspondence for Unicritical Polynomials

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell–Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton–Patel, and Vivaldi–Hatjispyros in the early 1990s connected the irreducibility and Galois-theoretic properties of dynatomic polynomials to rational periodic points; from the Galois–dynamics correspondence, we derive similar consequences for quadratic periodic points of unicritical polynomials. This is sufficient to deduce the non-existence of quadratic periodic points of quadratic polynomials with exact period 5 and 6, outside of a specified finite set from Morton and Krumm’s work in explicit Hilbert irreducibility.

A consequence of the relative Bogomolov conjecture

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 [DGH21].

A SPARSITY RESULT FOR THE DYNAMICAL MORDELL–LANG CONJECTURE IN POSITIVE CHARACTERISTIC

AbstractWe 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, a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map $\Phi :X{\longrightarrow } X$ defined over K, a point $\alpha \in X(K)$ and a subvariety $V\subseteq X$ , then the set of all nonnegative integers n such that $\Phi ^n(\alpha )\in V(K)$ is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X, $\Phi $ , $\alpha $ and V) such that for each positive integer M, $$\begin{align*}\scriptsize\#\{n\in S\colon n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align*}$$

Height gap conjectures, D-finiteness, and a weak dynamical Mordell–Lang conjecture

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 $f(\Phi^n(x))$, where $\Phi\colon X\to X$ 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 $\Phi$ and $f$. They conjectured that if the sequence is infinite, then $\limsup \frac{h(f(\Phi^n(x)))}{\log n} > 0$. They also made a corresponding conjecture for $\liminf$ and showed that it implies the Dynamical Mordell-Lang Conjecture. In this paper, we prove the $\limsup$ conjecture as well as the $\liminf$ conjecture away from a set of density $0$. As applications, we prove results concerning the growth rate of coefficients of $D$-finite power series as well as the Dynamical Mordell-Lang Conjecture up to a set of density $0$.

A Gap Principle for Subvarieties with Finitely Many Periodic Points

AbstractLet $f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety $X$ defined over a number field $K$ and fix an initial point $a\in X$. We consider a special case of the Dynamical Mordell–Lang Conjecture, where the subvariety $V$ contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set $\{n\in \mathbb{Z}_{{\geqslant}0}\mid f^{n}(a)\in V\}$ satisfies a strong gap principle.

Dynamical Uniform Bounds for Fibers and a Gap Conjecture

AbstractWe prove a uniform version of the Dynamical Mordell–Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our 1st result, we assume $X$ is a quasi-projective variety defined over a field $K$ of characteristic $0$, endowed with the action of an étale endomorphism $\Phi $, and $f\colon X\longrightarrow Y$ is a morphism with $Y$ a quasi-projective variety defined over $K$. Then for any $x\in X(K)$, if for each $y\in Y(K)$, the set $S_{x,y}:=\{n\in{\mathbb{N}}\colon f(\Phi ^n(x))=y\}$ is finite, then there exists a positive integer $N_x$ such that $\sharp S_{x,y}\le N_x$ for each $y\in Y(K)$. For our 2nd result, we let $K$ be a number field, $f:X\dashrightarrow{\mathbb{P}}^1$ is a rational map, and $\Phi $ is an arbitrary endomorphism of $X$. If ${\mathcal{O}}_{\Phi }(x)$ denotes the forward orbit of $x$ under the action of $\Phi $, then either $f({\mathcal{O}}_{\Phi }(x))$ is finite, or $\limsup _{n\to \infty } h(f(\Phi ^n(x)))/\log (n)>0$, where $h(\cdot )$ represents the usual logarithmic Weil height for algebraic points.

The dynamical Mordell-Lang problem for intersection of two orbits

Let X be an algebraic variety defined over C, let α∈X(C) and Φ:X⟶X be a morphism. The Dynamical Mordell-Lang conjecture describes the intersection of an orbit OrbΦ(α) with a closed subvariety of X. In this paper, we study the intersections of orbits in polynomial dynamics. In particular, various upper bounds are derived for the orbit intersection problem when X is an affine n-space and self maps are polynomial morphisms of special type.

THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

AbstractLet $K$ be an algebraically closed field of prime characteristic $p$, let $X$ be a semiabelian variety defined over a finite subfield of $K$, let $\unicode[STIX]{x1D6F7}:X\longrightarrow X$ be a regular self-map defined over $K$, let $V\subset X$ be a subvariety defined over $K$, and let $\unicode[STIX]{x1D6FC}\in X(K)$. The dynamical Mordell–Lang conjecture in characteristic $p$ predicts that the set $S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$ is a union of finitely many arithmetic progressions, along with finitely many $p$-sets, which are sets of the form $\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$ for some $m\in \mathbb{N}$, some rational numbers $c_{i}$ and some non-negative integers $k_{i}$. We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $X$ is an algebraic torus, we can prove the conjecture in two cases: either when $\dim (V)\leqslant 2$, or when no iterate of $\unicode[STIX]{x1D6F7}$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $X$. We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.

Strongly semistable sheaves and the Mordell–Lang conjecture over function fields

We give a new proof of the Mordell–Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer’s theorem that the Harder–Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. The interest of this proof is that it provides simple effective bounds (depending on the degree of the canonical line bundle) for the degree of the isotrivial finite cover whose existence is predicted by the Mordell–Lang conjecture. We also present a conjecture on the Harder–Narasimhan filtration of the cotangent bundle of a smooth projective variety of general type in positive characteristic and a conjectural refinement of the Bombieri–Lang conjecture in positive characteristic.

Powers in Orbits of Rational Functions: Cases of an Arithmetic Dynamical Mordell–Lang Conjecture

AbstractLet$K$be a finitely generated field of characteristic zero. For fixed$m\geqslant 2$, we study the rational functions$\unicode[STIX]{x1D719}$defined over$K$that have a$K$-orbit containing infinitely many distinct$m$-th powers. For$m\geqslant 5$we show that the only such functions are those of the form$cx^{j}(\unicode[STIX]{x1D713}(x))^{m}$with$\unicode[STIX]{x1D713}\in K(x)$, and for$m\leqslant 4$we show that 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$\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}$is a union of finitely many arithmetic progressions, where$\unicode[STIX]{x1D719}^{n}$denotes the$n$-th iterate of$\unicode[STIX]{x1D719}$and$\unicode[STIX]{x1D706}\in K(x)$is any map Möbius-conjugate over$K$to$x^{m}$. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell–Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves$y^{m}=\unicode[STIX]{x1D719}^{n}(x)$. We describe all$\unicode[STIX]{x1D719}$for which these curves have an irreducible component of genus at most 1, and show that such$\unicode[STIX]{x1D719}$must have two distinct iterates that are equal in$K(x)^{\ast }/K(x)^{\ast m}$.

Dynamical Mordell�Lang and automorphisms of blow-ups

We show that if $\phi : X \to X$ 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 $\phi^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X, \phi)$ 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 under an automorphism. These results follow from a non-reduced analogue of the dynamical Mordell-Lang conjecture. Namely, let $\phi : X \to X$ be an etale endomorphism of a smooth projective variety $X$ over a field $k$ of characteristic zero. We show that if $Y$ and $Z$ are two closed subschemes of $X$, then the set $A_\phi(Y,Z) = \{n : \phi^n(Y) \subseteq Z\}$ is the union of a finite set and finitely many residue classes, whose modulus is bounded in terms of the geometry of $Y$.