Points of bounded height on equivariant compactifications of vector groups, II

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

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

关于 $\mathbb{G}_a^n$ 的形式的等变紧化的 Manin 猜想

Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$

本文在边界除子满足一定条件时，证明了函数域上 $\mathbb{G}_a^n$ 的形式的光滑等变紧化的 Batyrev-Manin 猜想，并验证了首项常数与 Peyre 预测一致，同时研究了 $\mathbb{P}^{p-1}$ 情形以展示函数域中的新现象。

Height zeta functions of equivariant compactifications of unipotent groups

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

Height zeta functions of equivariant compactifications of the Heisenberg group

We study analytic properties of height zeta functions of equivariant compactifications of the Heisenberg group.

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 $\mathrm{PGL}_2$.

Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups

We apply the theory of height zeta functions to study the asymptotic distribution of rational points of bounded height on projective equivariant compactifications of semi-direct products.

几何Batyrev-Manin-Peyre理论对于加法群等价紧缩的渐进行为

Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups

本文研究了给定族在光滑射影曲线上的截面模空间的渐进行为，当通用纤维为有限维向量空间的等价紧缩时，证明了在合适的Grothendieck概形环中，这些模空间的类在某种归一化下收敛到非零有效元素，为Batyrev-Manin-Peyre猜想的动机版本提供了肯定答案。

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.

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.

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。

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。

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 $G := H^n$.

A group-invariant version of Lehmer's conjecture on heights

We state and prove an equivariant version of Lehmer's conjecture on heights, generalizing papers by Zagier (1993) and Dresden (1998) which are special cases of this theorem. We also extend their three cases to a full classification of all groups satisfying the condition that the set of all orbits for which every non-zero element lies on the unit circle is finite and non-empty.

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 …

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties

We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group endomorphisms of linear algebraic groups. We also propose a possible approach to the conjecture using equivariant MMP.

Rational points on certain homogeneous varieties

Let $L$ be a simply-connected simple connected algebraic group over a number field $F$, and $H$ be a semisimple absolutely maximal connected $F$-subgroup of $L$. Under a cohomological condition, we prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of $Δ(H)\backslash L^n$ with respect to a balanced line bundle, where $Δ(H)$ is the image of $H$ diagonally embedded in $L^n$.