Integral points of bounded height on compactifications of semi-simple groups

Height zeta functions of equivariant compactifications of unipotent groups

We prove Manin's conjecture for bi-equivariant compactifications of unipotent groups.

Distribution of rational points of bounded height on equivariant compactifications of $\mathrm{PGL}_2$ I

We study the distribution of rational points of bounded height on a one-sided equivariant compactification of $\mathrm{PGL}_2$ using automorphic representation theory of $\mathrm{PGL}_2$.

The Distribution of Integral Points on the Wonderful Compactification by Height

We study the asymptotic distribution of S-integral points of bounded height on partial bi-equivariant compactifications of semi-simple groups of adjoint type.

Rational points on compactifications of semi-simple algebraic groups

We prove Manin's conjecture concerning the distribution of rational points of bounded height, and its refinement by Peyre, for wonderful compactifications of semi-simple algebraic groups over number fields. The proof proceeds via the study of the associated height zeta function and its spectral expansion.

Campana points of bounded height on vector group compactifications

We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behaviour of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By i…

向量群紧化上有界高度的半整点

Semi-integral points of bounded height on vector group compactifications

本文用类似[PSTVA21]的思路得到向量群等变紧化上有界高度Darmon点计数函数的渐近公式，且该公式与[Moe26a]中$\boldsymbol{\textit{M}}$-点的Manin猜想相容。

An explicit semi-factorial compactification of the Néron model

C.~Pépin recently constructed a semi-factorial compactification of the Néron model of an abelian variety using the flattening technique of Raynaud--Gruson. Here we prove that an explicit semi-factorial compactification is a certain moduli space of sheaves --- the family of compactified jacobians.

齐次空间的整点渐近性、等变紧化与等分布

Asymptotics of integral points, equivariant compactifications and equidistributions for homogeneous spaces

针对线性代数群在Q上的齐次簇整点渐近计数问题，借助幂零流测度刚性将其约化为等变双有理几何问题，证明了特定条件下齐次空间的强Hardy-Littlewood性质及整点与体积渐近性的关系，并给出具体例子说明方法。

Equivariant compactifications of two-dimensional algebraic groups

We classify generically transitive actions of semidirect products of an additive and a multiplicative group on the projective plane. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for such semidirect products.

Semi-toric and toroidal compactifications as log minimal models, and applications to weak K-moduli

We give a characterization of toroidal (resp., semi-toric) compactifications due to Ash-Mumford-Rapoport-Tai (resp., Looijenga) as log minimal models and apply it to study weak K-moduli compactifications, giving a different proof to a theorem of Alexeev-Engel. We also discuss towards further generalization, in particular revisit Shah-Sterk compactification of moduli of polarized Enriques surfaces to show compatibility with log K-stability.

On heights of motives with semistable reduction

We study heights of motives with integral coefficients over number fields introduced by Kato. It is a generalization of the Faltings height of an abelian variety and we establish generalizations of some properties of the Faltings height in our context as conjectured by Kato. This sheds some light on integral structures of motives.

Geometrie d'Arakelov et hauteurs canoniques sur des varietes semi-abeliennes

Canonical heights and Arakelov geometry on semi-abelian varieties. In this paper, we propose a construction of the canonical heights on an extension of an abelian variety by the multiplicative group, in the framework of Arakelov geometry. These canonical heights are the sum of some height coming from the abelian variety and something we call a relative height. We finally give some complements about the points whose relative height is zero.

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of G_a^2, to assist with proofs of Manin's conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of a semidirect product of G_a and G_m.

环面簇上有界高度的半整点

Semi-integral points of bounded height on toric varieties

本文证明了环面簇上有界高度半整点的渐近公式，并在修正首项常数后验证了光滑及部分奇异环面轨形上的Manin型猜想。

优美紧化上的Campana点

Campana points on wonderful compactifications

本文证明了伴随型半单代数群优美紧化上Campana点的Manin猜想变体，并以此为关于该猜想首项常数的新猜想提供了证据。

Set Theory and p-adic Algebras

We verified that the existence of a maximal ideal of height 0 in a p-adic algebra in a certain class is independent of the axiom of ZFC. We established the theory on a P-point in the boundary of a topological space in the universal totally disconnected Hausdorff compactification. It is quite similar with the theory on a P-point in the boundary of a topological space in the Stone-Cech compactification. The latter theory relies on the real analysis, and the reason why the real analysis works for it is because the Stone-Cech compactification has the lifting property for a real bounded continuous …

Multiple Mixing for adele groups and rational points

We prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of $H\G$, where $H$ is a connected simple algebraic group embedded diagonally into $G := H^n$.

Height growth on semisimple groups

A condition is given, under which a general lattice point counting function is asymptotic to the corresponding ball volume growth function. This is then used to give height asymptotics in the style of the Batyrev-Manin Conjecture for certain intrinsically defined heights on semisimple groups.

Semistable models for modular curves and power operations for Morava E-theories of height 2

We construct an integral model for Lubin-Tate curves as moduli of finite subgroups of formal deformations over complete Noetherian local rings. They are p-adic completions of the modular curves X_0(p) at a mod-p supersingular point. Our model is semistable in the sense that the only singularities of its special fiber are normal crossings. Given this model, we obtain a uniform presentation for the Dyer-Lashof algebra of Morava E-theories at height 2 as local moduli of power operations in elliptic cohomology.

Compactifications of S-arithmetic quotients for the projective general linear group

Let F be a global field, and let S be a finite set of places of F containing all archimedean places. Consider the product X of the symmetric spaces and Bruhat-Tits buildings for PGL_d of the completions of F at archimedean and non-archimedean places in S, respectively. We construct compactifications of the quotient of X by S-arithmetic subgroups of PGL_d(F). The constructions make delicate use of reductive Borel-Serre spaces for archimedean places and polyhedral and seminorm compactifications at nonarchimedean places. We also briefly discuss a few potential applications of our compacifications…

The Bounded Height Conjecture for Semiabelian Varieties

The Bounded Height Conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in $G$. This conjecture has been shown by Habegger in the case where $G$ is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if $G$ is a general semiabelian variety. In particular, the lack of Poi…

Points of Small Height on Semiabelian Varieties

The Equidistribution Conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general case has remained intractable so far because the height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan's equidistribution theorem on algebraic dynamical systems. To overcome this, an asymptotic adaption of the equidistribution technique invented by Szpiro, Ullmo, and Zhang is used…

按高度计数半群轨道中的点

Counting points by height in semigroup orbits

针对极化动力系统半群轨道中高度有界点的计数问题，本文利用 Wiener-Ikerhara Tauberian 定理改进已有估计，并得到射影直线上一般半群的精确渐近式。

The geometric and arithmetic volume of Shimura varieties of orthogonal type

We apply the theory of Borcherds products to calculate arithmetic volumes (heights) of Shimura varieties of orthogonal type up to contributions from very bad primes. The approach is analogous to the well-known computation of their geometric volume by induction, using special cycles. A functorial theory of integral models of toroidal compactifications of those varieties and a theory of arithmetic Chern classes of integral automorphic vector bundles with singular metrics are used. We obtain some evidence in the direction of Kudla's conjectures on relations of heights of special cycles on these v…

仿射空间中模形的循环周期点及具有有界house和高度的周期点

Cyclotomic preperiodic points for morphisms in affine spaces and preperiodic points with bounded house and height

本文研究了仿射空间中模形的周期点集在循环闭包上的稠密性问题，并探讨了具有有界house和高度的周期点的性质。