Lang Conjecture Research Articles

Parshin’s method and the geometric Bombieri-Lang conjecture

In this short survey, we explain Parshin’s proof of the geometric Bombieri–Lang conjecture, and show that it can be used to give an alternative proof of Xie–Yuan’s recent resolution of the geometric Bombieri–Lang conjecture for projective varieties with empty special locus and admitting a finite morphism to a traceless abelian variety.

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.

Weakly Kähler hyperbolic manifolds and the Green–Griffiths–Lang conjecture

Abstract We introduce the notion of weakly Kähler hyperbolic manifold which generalizes that of Kähler hyperbolic manifold given in the early 1990s by M. Gromov, and establish its basic features. We then investigate its spectral properties and show a spectral gap result (on a suitable modification). As applications, we prove that weakly Kähler hyperbolic manifolds are of general type and we study the geometry of their subvarieties and entire curves, verifying – among other things – various aspects of the Lang and the Green–Griffiths conjectures for this class of manifolds.

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.

Dynamical Mordell–Lang conjecture for totally inseparable liftings of Frobenius

Dynamical Mordell–Lang conjecture for totally inseparable liftings of Frobenius

Degeneration locus of [formula omitted]-local systems: Conjectures

We introduce a conjecture on the arithmetic sparcity of the degeneration locus of a p-adic local system on a smooth variety over a number field and, modulo the Bombieri–Lang conjecture, show that it follows from a conjecture on the geometry of the level varieties attached to the local system. We present a few applications of our conjecture to classical problems in arithmetic geometry. Eventually, we give some evidences and discuss a few perspectives to attack it, in particular for p-adic local systems arising from geometry.

On the Geometric Zilber–Pink theorem and the Lawrence–Venkatesh method

Using our recent results on the algebraicity of the Hodge locus for variations of Hodge structures of level at least 3, we improve the results of Lawrence–Venkatesh in direction of the refined Bombieri–Lang conjecture.

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.

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.

Polynomial dynamics and local analysis of small and grand orbits

We prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role of the divisible group is taken by the small orbit of a point $\alpha$ where the small orbit by a polynomial $f$ is given by \begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*} Our main theorem is a classification of the algebraic relations that hold between infinitely many pairs of points in $\mathcal {S}_\alpha$ when everything is defined over the algebraic numbers and the degree $d$ of $f$ is at least 2. Our proof relies on a careful study of localisations of the dynamical system and follows an entirely different approach than previous proofs in this area. In particular, we introduce transcendence theory and Mahler functions into this field. Our methods also allow us to classify all algebraic relations that hold for infinitely many pairs of points in the grand orbit \begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*} of $\alpha$ if $|f^{\circ n}(\alpha )|_v \rightarrow \infty$ at a finite place $v$ of good reduction co-prime to $d$.

On the Bombieri–Lang conjecture over finitely generated fields

The strong Bombieri-Lang conjecture postulates that, for every variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, there exists an open subset $U\subset X$ such that $U(K)$ is finite for every finitely generated extension $K/k$. The weak Bombieri-Lang conjecture postulates that, for every positive dimensional variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, the rational points $X(k)$ are not dense. Furthermore, Lang conjectured that every variety of general type $X$ over a field of characteristic $0$ contains an open subset $U\subset X$ such that every subvariety of $U$ is of general type, this statement is usually called geometric Lang conjecture. We reduce the strong Bombieri-Lang conjecture to the case $k=\mathbb{Q}$. Assuming the geometric Lang conjecture, we reduce the weak Bombieri-Lang conjecture to $k=\mathbb{Q}$, too.

On the distribution of rational points on ramified covers of abelian varieties

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$, where $A$ is an abelian variety over $k$ with a dense set of $k$-rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $\pi (X(k))$ is disjoint from $C$. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the inverse Galois problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.

On the locus of higher order jets of entire curves in complex projective varieties

For a given complex projective variety, the existence of entire curves is strongly constrained by the positivity properties of the cotangent bundle. The Green–Griffiths–Lang conjecture stipulates that entire curves drawn on a variety of general type should all be contained in a proper algebraic subvariety. We present here new results on the existence of differential equations that strongly restrain the locus of entire curves in the general context of foliated or directed varieties, under appropriate positivity conditions.

A truncated second main theorem for algebraic tori with moving targets and applications

We establish a second main theorem for algebraic tori with slow growth moving targets with truncation to level 1. As the first application of this result, we prove the Green–Griffiths–Lang conjecture for projective spaces with n + 1 $n+1$ components in the context of moving targets of slow growth. Then we discuss the integrability of the ring of exponential polynomials in the ring of entire functions as another application.

Special holomorphic tensors on orthogonal modular varieties and applications to the Lang conjecture

We introduce a method to construct special holomorphic tensors on orthogonal modular varieties from scalar-valued modular forms, and give applications to the Lang conjecture on the birational type of subvarieties of orthogonal modular varieties.

Remark on the transcendence of real numbers generated by Thue–Morse along squares

In 1929, Mahler proved that the real number generated by Thue–Morse sequence is transcendental. Later, Adamczewski and Bugeaud gave a different proof of the transcendence of this number using a combinatorial transcendence criterion. Moreover, Kumar and Meher gave the generalization of the combinatorial transcendence criterion under the subspace Lang conjecture. In this paper, we prove under the subspace Lang conjecture that the real number generated by Thue–Morse along squares is transcendental by using the combinatorial transcendence criterion of Kumar and Meher.

Algebraic intermediate hyperbolicities

We extend Lang's conjectures to the setting of intermediate hyperbolicity and prove two new results motivated by these conjectures. More precisely, we first extend the notion of algebraic hyperbolicity (originally introduced by Demailly) to the setting of intermediate hyperbolicity and show that this property holds if the appropriate exterior power of the cotangent bundle is ample. Then, we prove that this intermediate algebraic hyperbolicity implies the finiteness of the group of birational automorphisms and of the set of surjective maps from a given projective variety. Our work answers the algebraic analogue of a question of Kobayashi on analytic hyperbolicity.

On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties (with an appendix by Dan Abramovich)

For maximal variational smooth families of projective manifolds whose general fibers have semi-ample canonical bundle, the Viehweg hyperbolicity conjecture states that the base spaces of such families are of log-general type. This deep conjecture was recently proved by Campana– Păun and was later generalized by Popa–Schnell. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log-general type. As a consequence, we prove the Brody hyperbolicity of moduli spaces of polarized manifolds with semi-ample canonical bundle. This proves a 2003 conjecture by Viehweg–Zuo.We also prove the Kobayashi hyperbolicity of base spaces for effectively parametrized families of minimal projective manifolds of general type. This generalizes previous work by To–Yeung, who further assumed that these families are canonically polarized.

Subvarieties of quotients of bounded symmetric domains

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each \(p \ge 1\), this criterion gives a precise condition under which the subvarieties \(V \subset X\) with \(\dim V \ge p\) are of general type, and X is p-measure hyperbolic. Then, we give several applications related to ball quotients, or to the Siegel moduli space of principally polarized abelian varieties. In the case of smooth compactifications of ball quotients, we obtain a conditional upper bound on the dimension on the exceptional locus, assuming the Green–Griffiths–Lang conjecture holds true. As another example of application, we give effective lower bounds for levels l so that the moduli spaces of genus g curves with l-level structures are of general type.