Zariski Density Research Articles

Integral points of bounded height on a certain toric variety

We determine an asymptotic formula for the number of integral points of bounded height on a certain toric variety, which is incompatible with part of a preprint by Chambert-Loir and Tschinkel. We provide an alternative interpretation of the asymptotic formula we get. To do so, we construct an analogue of Peyre’s constant α \alpha and describe its relation to a new obstruction to the Zariski density of integral points in certain regions of varieties.

Distance matrices of a tree: Two more invariants, and in a unified framework

A classical result of Graham and Pollak (1971) states that the determinant of the distance matrix DT of any tree T depends only on the number of edges of T. This and several other variants of DT have since been studied – including a q-version, a multiplicative version, and directed versions of these – and in all cases, det(DT) depends only on the edge-data.In this paper, we introduce a more general framework for bi-directed weighted trees that has not been studied to date; our work is significant for three reasons. First, our setting strictly generalizes – and unifies – all variants of DT studied to date (with coefficients in an arbitrary unital commutative ring) – including in Graham and Pollak (1971) above, as well as Graham and Lovász (1978), Yan and Yeh (2006), Yan and Yeh (2007), Sivasubramanian (2010), and others.Second, our results strictly improve on state-of-the-art for every variant of the distance matrix studied to date, even in the classical Graham–Pollak case. Here are three results for trees: (1) We compute the minors obtained by deleting arbitrary equinumerous sets of pendant nodes (in fact, more general sub-forests) from the rows and columns of DT, and show these minors depend only on the edge-data and not the tree-structure. (2) We compute a second function of the distance matrix DT: the sum of all its cofactors, termed cof(DT). We do so in our general setting and in stronger form, after deleting equinumerous pendant nodes (and more generally) as above – and show these quantities also depend only on the edge-data. (3) We compute in closed form the inverse of DT, extending a result of Graham and Lovász (1978) and answering an open question of Bapat et al. (2006) in greater generality.Third, a new technique is to crucially use commutative algebra arguments – specifically, Zariski density – which to our knowledge are hitherto unused for such matrices/invariants, but are richly rewarding. We also explain why our setting is “most general”, in that for more general edgeweights, det(DT),cof(DT) depend on the tree structure. In a sense, this completes the study of the invariants det(DT),cof(DT) for distance matrices of trees T with edge-data in a commutative ring.Our proofs use novel results for arbitrary bi-directed strongly connected graphs G: we prove a multiplicative analogue of an additive result by Graham et al. (1977), as well as a novel q-version thereof. In particular, we provide closed-form expressions for det(DG), cof(DG), and DG−1 in terms of their strong blocks. We then show how this subsumes the classical 1977 result, and provide sample applications to adding pendant trees and to cycle-clique graphs (including cactus/polycyclic graphs and hypertrees), subsuming variants in the literature. The final section introduces and computes a third – and novel – invariant for trees, as well as a parallel Graham–Hoffman–Hosoya type result for our “most general” distance matrix DT.

Zariski Density of Points with Maximal Arithmetic Degree

Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this paper, we show that there are densely many Q‾-rational points with maximal arithmetic degree (i.e., whose arithmetic degree is equal to the first dynamical degree) for self-morphisms on projective varieties. For unirational varieties and Abelian varieties, we show that there are densely many rational points with maximal arithmetic degree over a sufficiently large number field. We also give a generalization of a result of Kawaguchi and Silverman in the Appendix.

Zariski density of crystalline points

We show that crystalline points are Zariski dense in the deformation space of a representation of the absolute Galois group of a p-adic field. We also show that these points are dense in the subspace parameterizing deformations with the determinant equal to a fixed crystalline character. Our proof is purely local and works for all p-adic fields and all residual Galois representations.

Zariski density and computing with S-integral groups

We generalize our methodology for computing with Zariski dense subgroups of SL(n,Z) and Sp(n,Z), to accommodate input dense subgroups H of SL(n,Q) and Sp(n,Q). A key task, backgrounded by the Strong Approximation theorem, is computing a minimal congruence overgroup of H. Once we have this overgroup, we may describe all congruence quotients of H. The case n=2 receives particular attention.

Density of rational points on a family of del Pezzo surfaces of degree one

Let k be an infinite field of characteristic 0, and X a del Pezzo surface of degree d with at least one k-rational point. Various methods from algebraic geometry and arithmetic statistics have shown the Zariski density of the set X(k) of k-rational points in X for d≥2 (under an extra condition for d=2), but fail to work in generality when the degree of X is 1, leaving a large class of del Pezzo surfaces for which the question of density of rational points is still open. In this paper, we prove the Zariski density of X(k) when X has degree 1 and is represented in the weighted projective space P(2,3,1,1) with coordinates x,y,z,w by an equation of the form y2=x3+az6+bz3w3+cw6 for a,b,c∈k with a,c non-zero, under the condition that the elliptic surface obtained by blowing up the base point of the anticanonical linear system |−KX| contains a smooth fiber above a point in P1∖{(1:0),(0:1)} with positive rank over k. When k is of finite type over Q, this condition is sufficient and necessary.

Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

AbstractWe define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

Rational Points on Elliptic K3 Surfaces of Quadratic Twist Type

AbstractIn studying rational points on elliptic K3 surfaces of the form $$\begin{equation*} f(t)y^2=g(x), \end{equation*}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell–Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels–Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.

PATCHING AND THE COMPLETED HOMOLOGY OF LOCALLY SYMMETRIC SPACES

AbstractUnder an assumption on the existence of $p$-adic Galois representations, we carry out Taylor–Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated with $\operatorname{GL}_{n}$ over a number field. We use our construction, and some new results in non-commutative algebra, to show that standard conjectures on completed homology imply ‘big $R=\text{big}~\mathbb{T}$’ theorems in situations where one cannot hope to appeal to the Zariski density of classical points (in contrast to all previous results of this kind). In the case where $n=2$ and $p$ splits completely in the number field, we relate our construction to the $p$-adic local Langlands correspondence for $\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

Deformation Theory of the Trivial mod p Galois Representation for GLn

Abstract We study the rigid generic fiber $\mathcal{X}^\square _{\overline \rho }$ of the framed deformation space of the trivial representation $\overline \rho : G_K \to \textrm{GL}_n(k)$ where $k$ is a finite field of characteristic $p>0$ and $G_K$ is the absolute Galois group of a finite extension $K/\textbf{Q}_p$. Under some mild conditions on $K$ we prove that $\mathcal{X}^\square _{\overline \rho }$ is normal. When $p> n$ we describe its irreducible components and show Zariski density of its crystalline points.

Horizontal non-vanishing of Heegner points and toric periods

Let F be a totally real number field and A a modular GL2-type abelian variety over F. Let K/F be a CM quadratic extension. Let χ be a class group character over K such that the Rankin-Selberg convolution L(s,A,χ) is self-dual with root number −1. We show that the number of class group characters χ with bounded ramification such that L′(1,A,χ)≠0 increases with the absolute value of the discriminant of K.We also consider a rather general rank zero situation. Let π be a cuspidal cohomological automorphic representation over GL2(AF). Let χ be a Hecke character over K such that the Rankin–Selberg convolution L(s,π,χ) is self-dual with root number 1. We show that the number of Hecke characters χ with fixed ∞-type and bounded ramification such that L(1/2,π,χ)≠0 increases with the absolute value of the discriminant of K.The Gross–Zagier formula and the Waldspurger formula relate the question to horizontal non-vanishing of Heegner points and toric periods, respectively. For both situations, the strategy is geometric relying on the Zariski density of CM points on self-products of a quaternionic Shimura variety. The recent result [26,31,1] on the André–Oort conjecture is accordingly fundamental to the approach.

Large deviation principle for random matrix products

Under a Zariski density assumption, we extend the classical theorem of Cramer on large deviations of sums of i.i.d. real random variables to random matrix products.

The Hilbert Property for integral points of affine smooth cubic surfaces

TextIn this paper we prove that the set of S-integral points of the smooth cubic surfaces in A3 over a number field k is not thin, for suitable k and S. The proof relies on techniques developed by Corvaja and Zannier in the paper [8]. As a corollary, we obtain results on the complement in P2 of a smooth cubic curve, improving on Beukers' proof that the S-integral points are Zariski dense, for suitable S and k. With our method we reprove Zariski density, but our result is more powerful since it is a stronger form of Zariski density. We moreover prove that the rational integer points on the Fermat cubic surface x3+y3+z3=1 form a non-thin set and we link our methods to previous results of Lehmer, Miller–Woollett and Mordell. VideoFor a video summary of this paper, please visit https://youtu.be/DQnnbj1_-W8.

On automorphic points in polarized deformation rings

For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, Bockle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouvea-Mazur. We generalize Bockle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under assumptions necessary to apply a small $R = \mathbb{T}$ theorem and an assumption on the local mod $p$ representation, we prove that every irreducible component of the universal polarized deformation space contains an automorphic point. When combined with work of Chenevier, this implies new results on the Zariski density of automorphic points in polarized deformation space in dimension three.

Zariski density and computing in arithmetic groups

For $n > 2$, let $\Gamma$ denote either $SL(n, Z)$ or $Sp(n, Z)$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $H\leq \Gamma$. This forms the main component of our methods for computing with such arithmetic groups $H$. More generally, we provide algorithms for computing with Zariski dense groups in $\Gamma$. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.

Zariski Density of Monodromy Groups via a Picard–Lefschetz Type Formula

For the universal family of cyclic covers of projective spaces branched along hyperplane arrangements in general position, we consider its monodromy group acting on an eigen linear subspace of the middle cohomology of the fiber. We prove the monodromy group is Zariski dense in the corresponding linear group. It can be viewed as a degenerate analogy of Carlson-Toledo's result about the monodromy groups of smooth hypersurfaces [Duke Math. J. 97(3) (1999), 621-648]. The main ingredient in the proof is a Picard-Lefschetz type formula for a suitable degeneration of this family.

ANDRÉ–OORT CONJECTURE AND NONVANISHING OF CENTRAL -VALUES OVER HILBERT CLASS FIELDS

Let$F/\mathbf{Q}$be a totally real field and$K/F$a complex multiplication (CM) quadratic extension. Let$f$be a cuspidal Hilbert modular new form over$F$. Let${\it\lambda}$be a Hecke character over$K$such that the Rankin–Selberg convolution$f$with the${\it\theta}$-series associated with${\it\lambda}$is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg$L$-values$L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$, as${\it\chi}$varies over the class group characters of$K$. Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters${\it\chi}$such that$L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$increases with the absolute value of the discriminant of$K$. We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg$L$-values. The Waldspurger formula plays an underlying role.

Congruences and coordinate semirings of tropical varieties

In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence. Our main definition applies Zariski density to the algebraic structure of the coordinate semiring† of an affine supertropical algebraic set, which we tie to tropical geometry, especially in connection with the dimension of an affine variety, obtaining the analogs of classical results from dimension theory including catenarity. The second approach, based on the layered structure, is given in the appendix.

$$\mathbb {A}^1$$-connected varieties of rank one over nonclosed fields

In this paper, we proved two results regarding the arithmetics of separably $\mathbb{A}^1$-connected varieties of rank one. First we proved over a large field, there is an $\mathbb{A}^1$-curve through any rational point of the boundary, if the boundary divisor is smooth and separably rationally connected. Secondly, we generalize a theorem of Hassett-Tschinkel for the Zariski density of integral points over function fields of curves.

Transience of algebraic varieties in linear groups - applications to generic Zariski density

We study the transience of algebraic varieties in linear groups. In particular, we show that a “non elementary” random walk in SL 2 (ℝ) escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk takes place in the real points of a semisimple split algebraic group and show such a result for a wide family of random walks.