These notes grew out of a mini-course given from May 13th to May 17th at UQAM in Montreal during a workshop on Diophantine Approximation and Value Distribution Theory. We start with an overview of Lang-Vojta's conjectures on pseudo-hyperbolic projective varieties. These conjectures relate various different notions of hyperbolicity. We begin with Brody hyperbolicity and discuss conjecturally related notions of hyperbolicity in arithmetic geometry and algebraic geometry in subsequent sections. We slowly work our way towards the most general version of Lang-Vojta's conjectures and provide a summary of all the conjectures, as well as a survey of recent progress.
Assuming Vojta's conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and positive integer $g$, there is an integer $m_0$ such that for any $m > m_0$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta's conjecture for Deligne-Mumford stacks, which we deduce from Vojta's conjecture for schemes.
We formulate a generalization of Vojta's conjecture in terms of log pairs and variants of multiplier ideals. In this generalization, a variety is allowed to have singularities. It turns out that the generalized conjecture for a log pair is equivalent to the original conjecture applied to a log resolution of the pair. A special case of the generalized conjecture can be interpreted as representing a general phenomenon that there tend to exist more rational points near singular points than near smooth points. The same phenomenon is also observed in relation between greatest common divisors of int…
We apply Vojta's conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta's conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.
Vojta's Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic varieties with nonnegative Kodaira dimension. We use this to deduce, from Vojta's Conjectures, conjectures of Batyrev-Manin and Manin on the distribution of rational points on algebraic varieties. In particular, we show that Vojta's Main Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces.
We introduce a qualitative conjecture, in the spirit of Campana, to the effect that certain subsets of rational points on a variety over a number field, or a Deligne-Mumford stack over a ring of S-integers, cannot be Zariski dense. The conjecture interpolates, in a way that we make precise, between Lang's conjecture for rational points on varieties of general type over number fields, and the conjecture of Lang and Vojta that asserts that S-integral points on a variety of logarithmic general type are not Zariski-dense. We show our conjecture follows from Vojta's conjecture. Assuming our conject…
We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $\mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta's Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan's Second Main Theorem that has been obtained by M. Ru and P. Vojta.
We prove new cases of Vojta's conjectures for surfaces in the context of function fields, with truncation equal to one and providing an effective explicit description of the exceptional set. We also prove a general and explicit result towards Campana's conjecture over complex function fields of curves. Our methods rely on a local study of $ω$-integral varieties.
We construct a family of fibered threefolds $X_m \to (S , Δ)$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,Δ)$ of general type. Following Campana, the threefolds $X_m$ are called \emph{weakly special} but not \emph{special}. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if $m$ is big enough the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conject…
We show that, assuming Vojta's height conjecture, the height of a rational point on an algebraically hyperbolic variety can be bounded "uniformly" in families. This generalizes a result of Su-Ion Ih for curves of genus at least two to higher-dimensional varieties. As an application, we show that, assuming Vojta's height conjecture, the height of a rational point on a curve of general type is uniformly bounded. Finally, we prove a similar result for smooth hyperbolic surfaces with $c_1^2 > c_2$.
We prove the nonsplit case of the Lang-Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $\mathbb{G}_m^2$. This extends results of Corvaja and Zannier, who proved the conjecture in the split case, and results of Corvaja and Zannier and the second author that were obtained in the case of the complement of a degree four and three component divisor in $\mathbb{P}^2$. We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.
Using the techniques introduced by Corvaja and Zannier we solve the non-split case of the geometric Lang-Vojta Conjecture for affine surfaces isomorphic to the complement of a conic and two lines in the projective plane. In this situation we deal with sections of an affine threefold fibered over a curve, whose boundary, in the natural projective completion, is a quartic bundle over the base whose fibers have three irreducible components. We prove that the image of each section has bounded degree in terms of the Euler Characteristic of the base curve.
In this short note we discuss the exceptional locus for the Lang-Vojta's conjecture in the case of the complement of two completely reducible hyperplane sections in a cubic surface. Using elementary methods, we show that generically the exceptional set is the union of the remaining 21 lines in the surface. We also describe examples in which the exceptional set is strictly larger.
We apply Schmidt's Subspace Theorem to establish Arithmetic General Theorems for projective varieties over number and function fields. Our first result extends an analogous result of M. Ru and P. Vojta. One aspect to its proof makes use of a filtration construction which appears in work of Autissier. Further, we consider work of M. Ru and J. T.-Y. Wang, which pertains to an extension of K. F. Roth's theorem for projective varieties in the sense of D. McKinnon and M. Roth. Motivated by these works, we establish our second Arithmetic General Theorem, namely a form of Roth's theorem for exception…
Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta's conjecture. More generally, we show that if $X$ is a $\mathbb{Q}$-factorial terminal split toric variety of arbitrary dimension, then $P$ is better approximated by points on a rational curve than by any Zariski dense sequence.
We show that the Vojta (or Hall-Lang) conjecture implies that the arboreal Galois representations in a 1-parameter family of quadratic polynomials are surjective if and only if they surject onto some finite and uniform quotient. As an application, we use the Vojta conjecture, our uniformity theorem over $\mathbb{Q}(t)$, and Hilbert's irreducibility theorem to prove that the prime divisors of many quadratic orbits have density zero.
In this paper, we consider the limit $ \lim_{n \to \infty} \sum_{v\in S} λ_{Y,v}(f^{n}(x))/h_{H}(f^{n}(x)) $ where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$ over a number field, $S$ is a finite set of places, $ λ_{Y,v}$ is a local height function associated with a proper closed subscheme $Y \subset X$, and $h_{H}$ is an ample height function on $X$. We give a geometric condition which ensures that the limit is zero, unconditionally when $\dim Y=0$ and assuming Vojta's conjecture when $\dim Y\geq1$. In particular, we prove (one is unconditio…
We investigate versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings, for other classes of varieties. We first obtain analogues for certain Fano threefolds. We use these results to prove the Shafarevich conjecture for smooth sextic surfaces, which appears to be the first non-trivial result in the literature on the arithmetic of such surfaces. Moreover, we exhibit certain moduli stacks of Fano varieties which are not hyperbolic, which allows us to show that the analogue of the Shafarevich conjecture does not always hold for Fano varieties. Our results al…
Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties…
Assuming Vojta's conjecture, we give a sufficient condition for the limit \[ \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x))} \] is equal to zero, where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$, $h_{H}$ is an ample height function on $X$, and $h_{Y}$ is a global height function associated with a closed subscheme $Y \subset X$ of codimension at least two. Based on this, we propose a conjecture on a sufficient condition for the limit to be zero. We point out that our conjecture implies Dynamical Mordell-Lang conjecture for endomor…
Motivated by Lang-Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov-Shramov's notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again Amerik's theorem and Prokhorov-Shramov's notion of quasi-minimal model, but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furth…
A common practice in arithmetic geometry is that of generalizing rational points on projective varieties to integral points on quasi-projective varieties. Following this practice, we demonstrate an analogue of a result of L. Caporaso, J. Harris and B. Mazur, showing that the Lang - Vojta conjecture implies a uniform bound on the number of stably integral points on an elliptic curve over a number field, as well as the uniform boundedness conjecture (Merel's theorem).
A widely believed conjecture predicts that curves of bounded geometric genus lying on a variety of general type form a bounded family. One may even ask whether the canonical degree of a curve $C$ in a variety of general type is bounded from above by some expression $aχ(C)+b$, where $a$ and $b$ are positive constants, with the possible exceptions corresponding to curves lying in a strict closed subset (depending on $a$ and $b$). A theorem of Miyaoka proves this for smooth curves in minimal surfaces, with $a>3/2$. A conjecture of Vojta claims in essence that any constant $a>1$ is possible provid…
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 $\mathbb{G}_a$.
We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of PEL-type.
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…
Assuming Lang's conjecture, we prove that for a fixed prime $p$, number field $K$, and positive integer $g$, there is an integer $r$ such that no principally polarized abelian variety $A/K$ of dimension $g$ has full level $p^r$ structure. To this end, we use a result of Zuo to prove that for each closed subvariety $X$ in the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$, there exists a level $m_X$ such that the irreducible components of the preimage of $X$ in $\mathcal{A}_g^{[m]}$ are of general type for $m > m_X$.
Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation
Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry and arithmetic of curves Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's and Campana's conjectures. Lecture 2: Campana's program; Campana constellations framed in terms of b-divisors, to allow for a definition of Kodaira dimension directly on the base. A speculative notion of firmaments which may deserve further investigation, espec…
In this paper we prove that given a pair $(X,D)$ of a threefold $X$ and a boundary divisor $D$ with mild singularities, if $(K_X+D)$ is movable, then the orbifold second Chern class $c_2$ of $(X,D)$ is pseudo-effective. This generalizes the classical result of Miyaoka on the pseudo-effectivity of $c_2$ for minimal models. As an application we give a simple solution to Kawamata's effective non-vanishing conjecture in dimension $3$, where we prove that $H^0(X, K_X+H)\neq 0$, whenever $K_X+H$ is nef and $H$ is an ample, effective, reduced Cartier divisor. Furthermore, we study Lang-Vojta's conjec…
We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine as well as projective varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and étale cohomology spaces of projective varieties. The gen…
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 prove the isogeny property for special fibres of integral canonical models of compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type. The approach used also shows that many crystalline cycles on abelian varieties over finite fields which are specializations of Hodge cycles, are algebraic. These two results have many applications. First, we prove a variant of the conditional Langlands--Rapoport conjecture for these special fibres. Second, for certain isogeny sets we prove a variant of the unconditional Langlands--Rapoport conjecture (like for many basic loci). Third, we prov…
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.
Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twisted Hecke correspondence acting on the compactly supported étale cohomology and verify another conjecture of Kottwitz \cite{Kottwitz90} on the stabilization of that formula. The main ingredients of our proof of the formula are a recent work of Kisin \cite{Kisin17} on Langlands-Rapoport conjecture and the theory of Galois gerbs developed by Langlands and Rapoport \cite{LR87}. Es…
Let $A$ be a non-zero abelian variety over a field $F$ that is not algebraic over a finite field. We prove that the rational rank of the abelian group $A(F)$ is infinite when $F$ is large in the sense of Pop (also called ample). The main ingredient is a deduction of the 1-dimensional case of the relative Mordell-Lang conjecture from a result of Rössler.
This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.
