Height zeta functions of equivariant compactifications of the Heisenberg group

Fano threefolds as equivariant compactifications of the vector group

In this article, we determine all equivariant compactifications of the three-dimensional vector group $\mathbf{G}_a^3$ which are smooth Fano threefolds with Picard number greater or equal than two.

生成全实域zeta值的典范等变上同调类

Canonical Equivariant Cohomology Classes Generating Zeta Values of Totally Real Fields

本文将Dirichlet $L$-函数特殊值的经典结果推广到全实域Hecke $L$-函数，在代数环面的等变上同调中构造Shintani生成类，证明其导数在挠点处的特殊化给出Lerch型zeta函数在非正整数处的值，表明高维情形的正确框架应为等变上同调类而非函数。

On the distribution of points of bounded height on equivariant compactifications of vector groups

We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space. (Nous établissons un développement asymptotique du nombre de points rationnels de hauteur bornée sur les compactifications équivariantes lisses de l'espace affine.)

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

We prove asymptotic formulas for the number of rational points of bounded height on certain blow-ups of the projective space.

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.

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 $\mathrm{PGL}_2$.

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。

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.

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。

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。

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.

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.

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。

A class of non-holomorphic modular forms II : equivariant iterated Eisenstein integrals

We introduce a new family of real analytic modular forms on the upper half plane. They are arguably the simplest class of `mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q, \overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first non-trivial functions in this class are real analytic Eis…

On zeta elements for $\mathbb{G}_{m}$

In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of `zeta element', of `Weil-étale cohomology complexes' and of `integral Selmer groups' for the multiplicative group $\mathbb{G}_m$ over finite abelian extensions of number fields. We then conjecture a precise connection between zeta elements and Weil-étale cohomology complexes, we show this conjecture is equivalent to a special case of the equivariant Tamagawa number conjecture and we give an unconditional proof of the analogous statement for global function fie…

Regularization and generalized double shuffle relations for p-adic multiple zeta values

We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.

An Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications

We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a $G$-equivariant $L$-function $Θ_{K/F}^E$ and prove an equivariant Tamagawa number formula for certain Euler-completed versions of its special value $Θ_{K/F}^E(0)$. This generalizes Taelman's class number formula for the value $ζ_F^E(0)$ of the Goss zeta function $ζ_F^E$ associated to the pair $(F, E)$. Taelman's result is obtained from our result by setting $…

从欧拉积分表示出发的超几何动机

Hypergeometric Motives from Euler Integral Representations

本文通过欧拉积分表示研究超几何函数的有理函数覆盖族，提出局部紧致化方法，并利用L函数和环面ζ函数的乘积形式，发展出计算希格拉特数的组合算法。

等变代数K理论与Artin L-函数

Equivariant algebraic $\mathrm{K}$-theory and Artin $L$-functions

本文将Quillen-Lichtenbaum猜想推广到有限域、函数域和全实代数数域的Galois表示的Artin L-函数，利用等变谱序列和Möbius反演公式，在许多情形下整系数地证明了该猜想并确定了有限域情形下的等变代数K群结构。

Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory upon integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet'…

Spectral symmetries of zeta functions

We define, answering a question of Sarnak in his letter to Bombieri, a symplectic pairing on the spectral interpretation (due to Connes and Meyer) of the zeroes of Riemann's zeta function. This pairing gives a purely spectral formulation of the proof of the functional equation due to Tate, Weil and Iwasawa, which, in the case of a curve over a finite field, corresponds to the usual geometric proof by the use of the Frobenius-equivariant Poincaré duality pairing in etale cohomology. We give another example of a similar construction in the case of the spectral interpretation of the zeroes of a c…

Cohomology of GL(4,Z) with Non-trivial Coefficients

In this paper we compute the cohomology groups of GL(4,Z) with coefficients in symmetric powers of the standard representation twisted by the determinant. This problem arises in Goncharov's approach to the study of motivic multiple zeta values of depth 4. The techniques that we use include Kostant's formula for cohomology groups of nilpotent Lie subalgebras of a reductive Lie algebra, Borel-Serre compactification, a result of Harder on Eisenstein cohomology. Finally, we need to show that the ghost class, which is present in the cohomology of the boundary of the Borel-Serre compactification, di…

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…

Hurwitz 空间、Nichols 代数与 Igusa zeta 函数

Hurwitz spaces, Nichols algebras, and Igusa zeta functions

本文构造次数 4、5 的 Hurwitz 空间拟映射紧化，建立算术统计、Nichols 代数与 Igusa zeta 函数的精确联系，并回答相关计数与拓扑起源问题。

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 $G := H^n$.

特征 $p$ 下可构造层的 Weil-étale 上同调与等变玉川数猜想

Weil-étale cohomology and the equivariant Tamagawa number conjecture for constructible sheaves in characteristic $p$

本文针对有限域上簇的可构造 étale 层，利用 Weil-étale 上同调证明了相关非交换 L-函数在负整数处的特殊值公式，从而给出等变玉川数猜想的几何类比并推广了若干已有结果。

具有$\u2124_2$对称性的Calabi-Yau流形的通量真空与模性

Flux Vacua and Modularity for $\mathbb{Z}_2$ Symmetric Calabi-Yau Manifolds

本研究在具有$\u2124_2$对称性的多参数Calabi-Yau流形的IIB紧致化中找到超对称通量真空连续族，通过扩展方法计算局部zeta函数，验证其分子二次因子对应权2模形式，支持通量模性猜想，并拓展至代数数域参数与相关模形式的研究。

关于 Artin $L$-函数的 Borel 秩定理

Borel's rank theorem for Artin $L$-functions

本文推广 Borel 秩定理，利用带有理 Galois 表示系数的等变代数 $K$-群及 Moore 谱扭变，建立 Artin $L$-函数版本，并探讨整系数情形的应用。