A. N. Parshin
We discuss the following topics: n-dimensional local fields and adelic groups; harmonic analysis on local fields and adelic groups for two-dimensional schemes (function spaces, Fourier transform, Poisson formula); representations of discrete Heisenberg groups; adelic Heisenberg groups and their representations arising from two-dimensional schemes.
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We establish an arithmetic intersection theory in the framework of Arakelov geometry over adelic curves. To each projective scheme over an adelic curve, we associate a multi-homogenous form on the group of adelic Cartier divisors, which can be written as an integral of local intersection numbers along the adelic curve. The integrability of the local intersection number is justified by using the theory of resultants.
Adelic Line Bundles, Arithmetic Positivity and Diophantine Geometry
本文综述拟射影簇上 adelic 线丛的基础、算术正性及其在均匀分布定理和一致 Bogomolov 猜想中的应用。
Arithmetic Fujita approximation over adelic curves
本文在adelic曲线的Arakelov理论框架下，证明了Fujita近似定理的类比结果，验证了Chen Huayi和Moriwaki Atsushi的猜想。
We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact sequence. We relate the last term of this short exact sequence with the projective limit of groups which are finite direct products of copies of one-dimensional real torus and are connected with first cohomology groups of locally free sheaves on the arithmetic surface.
We establish, in the setting of Arakelov geometry over adelic curves, an arithmetic Hilbert-Samuel theorem describing the asymptotic behaviour of the metrized graded linear series of an adelic line bundle in terms of its arithmetic intersection number. We then study positivity conditions of adelic line bundles.
We work with completed adelic structures on an arithmetic surface and justify that the construction under consideration is compatible with Arakelov geometry. The ring of completed adeles is algebraically and topologically self-dual and fundamental adelic subspaces are self orthogonal with respect to a natural differential pairing. We show that the Arakelov intersection pairing can be lifted to an idelic intersection pairing.
This is the first paper of a series. We prove an arithmetic Hodge index theorem for adelic line bundles on projective varieties over number fields. It extends the arithmetic Hodge index theorem of Faltings, Hriljac and Moriwaki on arithmetic varieties. As consequences, we obtain the uniqueness part of the non-archimedean Calabi--Yau theorem, and a rigidity property of the sets of preperiodic points of polarizable algebraic dynamical systems over number fields.
In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic $\mathbb{R}$-Cartier divisor $\overline{D}$ are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized $\mathbb{R}$-divisors on toric varie…
We give an introduction to adelic mixing and its applications for mathematicians knowing about the mixing of the geodesic flow on hyperbolic surfaces. We focus on the example of the Hecke trees in the modular surface.
This is the second paper of a series. It extends the results of the first paper from number fields to finitely generated fields, based on the recent theory of adelic line bundles of the same authors. We prove an arithmetic Hodge index theorem for these adelic line bundles, and apply the theorem to obtain a rigidity property of the sets of preperiodic points of polarizable algebraic dynamical systems over any field.
This paper is the second part of arXiv:0707.1766. We develope harmonic analysis in some categories of filtered abelian groups and vector spaces over the fields R or C. These categories contain as objects local fields and adelic spaces arising from arithmetical surfaces. Some structure theorems are proven for quotients of the adelic groups of algebraic and arithmetical surfaces.
Global pluripotential theory for adelic line bundles
本文将拟射影算术簇上的adelic线丛理论与整体Berkovich空间的多重位势理论联系起来，建立了adelic线丛子范畴与解析化上配备连续多次下调和度量的线丛的等价关系，并推广了Monge-Ampère测度的构造与应用。
Abstract divisorial spaces and arithmetic intersection numbers
本文引入抽象除数空间，用于推广算术交数至Chen和Moriwaki意义下的proper adelic基曲线，同时允许非阿基米德地方使用相对混合能量的更奇异度量。
Baker-Rumely and Favre-Rivera-Letelier independently proved an important arithmetic equidistribution theorem for points of small height on the Berkovich compactification of the projective line with respect to an adelic measure on $\mathbb{P}^1$. Around the same time, Chambert-Loir proved a more general version of this arithmetic equidistribution theorem in the setting of curves from a different approach. We generalize the notion of an adelic measure to that of a quasi-adelic measure on $\mathbb{P}^1$, and show that arithmetic equidistribution of points with small height holds for quasi-adelic …
Arithmetic bigness and a uniform Bogomolov-type result
本文证明了曲线泛族的容许典范丛是大的adelic线丛，并基于此证明了所有特征global fields上曲线的一致Bogomolov型定理，为一致Mordell-Lang型结果提供了新方法。
We use a form of lifted harmonic analysis to develop a two-dimensional adelic integral representation of the zeta functions of simple arithmetic surfaces. Manipulations of this integral then lead to an adelic interpretation of the so-called mean-periodicity correspondence, which is comparable to the better known automorphicity conjectures for the generic fibre.
Heights on toric varieties for singular metrics: Global theory
本文提出了一种极核种的极值理论，通过凹函数积分计算极核算术品种的极值。
On a relation between Deninger's foliated dynamical systems and Connes-Consani's adelic spaces
本文探讨了Deninger的叶状动力系统与Connes-Consani的adelic空间之间的关系，揭示了类域论的几何视角。
We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic $1$-dimensional $\mathbb{K}$-lattices (here and subsequently, $\mathbb{K}$ denotes a quadratic imaginary number field) and define theta functions on it. We also embed the groupoid of commensurability modulo dilations into the union of adelic Heisenbe…
This work introduces adelic constructions of direct images of differentials and symbols in the two-dimensional case in the relative situation. In particular, reciprocity laws for relative residues of differentials and symbols are stated and applied to a construction of the Gysin map for Chow groups.
Arithmetic Demailly Approximation Theorem
本文将Demailly逼近定理从复几何推广到Arakelov几何，并利用该结果在无需相对半正定假设下证明了adelic线丛的本质非负性蕴含伪有效性，同时将结论推广到拟射影簇情形。
We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck's trace map of the surface as a sum of residues. Points at infinity are then incorporated into the theory and the reciprocity law is extended to all curves on the surface. Applications to adelic duality for the arithmetic surface are discussed.
On the height of the universal abelian variety
本文推广袁-X和张-S.W.的算术相交理论至非nef且不可积的adelic算术除子，用相对有限能量概念证明泛阿贝尔簇上Siegel-Jacobi形式线丛不可积并算出其算术自交数，技术可用于混合Shimura簇等场景。
Construction of Arithmetic Teichmuller Spaces I
本文引入算术全纯结构，分别构造了p进域和数域上簇的算术泰希米勒空间，拓展了数域与黎曼曲面的类比，思路不同于望月新一的相关理论。
In this note, we establish a duality result under the residue paring between certain two-dimensional adelic spaces, which are associated to a closed point on an arithmetic surface.
In this paper we give a simple, explicit construction of polytopal bounded remainder sets of all possible volumes, for any irrational rotation on the $d$ dimensional adelic torus $\mathbb{A}^d/\mathbb{Q}^d$. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument that reduces the existence argument to the case of an irrational rotation on the torus $\mathbb{R}^d/\mathbb{Q}^d$.
A probabilistic interpretation of Weil's explicit sums and arithmetic spectral measures
本文通过引入实数域的 Bohr 紧化作为概率空间，将 Weil 显式公式表达为随机变量协方差与期望值及谱积分极限，从而给出其概率与几何解释。
Equidistribution for semiabelian varieties over number fields
本文针对数域上半阿贝尔簇，在环面赋值为单临界且算术T-有效时，证明了一般等分布，建立了不同类型环面赋值的比较机制，进而得到对应Bogomolov定理。
We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings and apply our criteria to study the effect of abstract field automorphisms of $\mathbb{C}$ on modular embeddings. Finally we derive that the absolute Galois group of $\mathbb{Q}$ operates on the dessins d'enfants defined by principal congruence subgroups of (arithmetic or non-ar…
This work describes several first steps in extending Tate-Iwasawa's analytic method to define an L-function in higher dimensions. For generalizing this method the author advocates the usefulness of the classical Riemann-Hecke approach, his adelic complexes together with his generalization of Krichever's correspondence. He analyzes dimension 1 types of functions and discusses properties of the lattice of commensurable classes of subspaces in the adelic space associated to a divisor on an algebraic surface.
The reduced norm-one group G of a central simple algebra is an inner form of the special linear group, and an involution on the algebra induces an automorphism of G. We study the action of such automorphisms in the cohomology of arithmetic subgroups of G. The main result is a precise formula for Lefschetz numbers of automorphisms induced by involutions of symplectic type. Our approach is based on a careful study of the smoothness properties of group schemes associated with orders in central simple algebras. Along the way we also derive an adelic reformulation of Harder's Gauss-Bonnet Theorem.
We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adelic torus and are closely related to class groups of number fields. The distribut…
We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P^1. Our set-up provides an extension of several aspects of the classical Fr"ohlich theory of the Galois module structure of rings of integers of …
On the generalisation of Roth's theorem
本文在adelic曲线上推广Roth定理，提出两种推广并证明收紧假设可增强不等式，恢复Corvaja和Vojta的已知结果。
Let $ϕ$ be a non-isotrivial family of Drinfeld A-modules of rank r in generic characteristic with a suitable level structure over a connected smooth algebraic variety X. Suppose that the endomorphism ring of $ϕ$ is equal to A. Then we show that the closure of the analytic fundamental group of X in $SL_r(\mathbb{A}_F^f)$ is open, where $\mathbb{A}_F^f$ denotes the ring of finite adeles of the quotient field F of A. From this we deduce two further results: (1) If X is defined over a finitely generated field extension of F, the image of the arithmetic étale fundamental group of X on the adelic Ta…
In this paper we will give the calculus, the criterion, and the existence of the arithmetic Galois covers of higher relative dimensions.
Selective Integration on Higher Adeles and the Euler Characteristic of Surfaces
本文提出了一种基于选择性子商的二维几何 Adeles 上的翻译不变测度和积分方法，用于计算曲面的欧拉特征及其交数。
The Arithmetic of the Fourier Coefficients of Automorphic Forms
本论文从经典与阿代尔视角研究模形式，推导其傅里叶系数的算术性质：计算带非平凡中心特征的局部惠特克新形式的$p$进赋值下界，刻画希尔伯特新形式在任意尖点处傅里叶系数的生成域，并讨论两部分关联与未来方向。
Rigidity of periodic points for loxodromic automorphisms of affine surfaces
本文证明仿射曲面上动力学次数大于1的自同构若共享稠密周期点集则周期点完全相同，通过构造典范高度并利用算术等分布理论（含Moriwaki高度情形）完成证明。
Harmonic analysis approach to the relative Riemann-Roch theorem on global fields
本文以纯 adelic 语言表示相对 Euler 示性数，通过规范化 adeles 上的 Haar 测度，从 adelic Poisson 求和公式推导出相对 Riemann-Roch 定理，并定义了相对上同调大小，统一覆盖了绝对与相对等多种情形。
We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over a number field, which is similar to Gillet and Soulé's definition of arithmetic Chow groups. We also give a compact description of the intersection theory of such groups. A consequence of this theory is the definition of a height pairing between two higher algebraic cycles, of complementary dimensions, whose real regulator class is zero. This description agrees with Beilinson's height pairing for the classical arithmetic Chow groups. We also give examples of the higher arithmetic intersection…
We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field $\mathbb{F}_q$. Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the $\ell$-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree $n$ in the limit as $q \to \infty$. The results of this paper can be v…
Suppose that $f \colon X \dashrightarrow X$ is a dominant rational self-map of a smooth projective variety defined over ${\overline{\mathbf Q}}$. Kawaguchi and Silverman conjectured that if $P \in X({\overline{\mathbf Q}})$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $λ_1(f)$ of $f$ if the orbit of $P$ is Zariski dense in $X$. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $X$. We begin by defining a set of arithmeti…
Higher order cohomology of arithmetic groups is expressed in terms of (g,K)-cohomology. Generalizing results of Borel, it is shown that the latter can be computed using functions of (uniform) moderate growth. A higher order versions of Borel's conjecture is stated, asserting that the cohomology can be computed using automorphic forms.
We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree two and higher in $\mathbb{F}_q[t]$, in the limit as $q\to\infty$. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-one $L$-functions (i.e. situations considered previously using this approach). They correspond to expressions found recently in the…
Divisibility and torsion in higher Chow groups over arithmetic fields
本文研究算术域上光滑概形的高阶Chow群的可除性与无挠性，并应用于有限域、局部域和整体域。
We give here a simple proof of the centrality of the congruence subgroup kernel in the higher rank isotropic case.
The notion of a tamely ramified covering is canonical only for curves. Several notions of tameness for coverings of higher dimensional schemes have been used in the literature. We show that all these definitions are essentially equivalent. Furthermore, we prove finiteness theorems for the tame fundamental groups of arithmetic schemes.
In this paper, we would like to propose a fundamental question about a higher dimensional analogue of Dirichlet's unit theorem. We also give a partial answer to the question as an application of the arithmetic Hodge index theorem.
Residues to a given modulus have been introduced to mathematics by Carl Friedrich Gauss with the definition of congruence in the `Disquisitiones Arithmeticae'. Their extraordinary properties provide the basis for a change of paradigm in arithmetic. By restricting residues to remainders left over by divison Peter Gustav Lejeune Dirichlet - Gauss's successor in Göttingen - eliminated in his `Lectures on number theory' the fertile concept of residues and attributed with the number-theoretic approach to residues for more than one and a half centuries to obscure Gauss's paradigm shift in mathematic…
A high-codimensional Yuan's inequality and its application to higher arithmetic degrees
本文推广袁氏算术Siu不等式至高余维，证明高维算术度存在性并建立相对度公式，给出Dang等人提出的广义Kawaguchi-Silverman猜想的反例。
Etale descent obstruction and anabelian geometry of curves over finite fields
本文通过建立平展基本群之间的态射与全局函数域上曲线的局部常值阿代尔点之间的双射，将Grothendieck的阿纳贝尔猜想与Sutherland和第二个作者的最新猜想联系起来，为阿纳贝尔猜想提供了进一步证据。
Let $Γ$ be a centerless irreducible higher rank arithmetic lattice in characteristic zero. We prove that if $Γ$ is either non-uniform or is uniform of orthogonal type and dimension at least 9, then $Γ$ is bi-interpretable with the ring $\mathbb{Z}$ of integers. It follows that the first order theory of $Γ$ is undecidable, that all finitely generated subgroups of $Γ$ are definable, and that $Γ$ is characterized by a single first order sentence among all finitely generated groups.
We study the density of rational points on a higher-dimensional orbifold $(\mathbb{P}^{n-1},D)$ when $D$ is a $\mathbb{Q}$-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy-Littlewood circle method to first study an asymptotic version of Waring's problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov's mean value theorem, due to Bourgain-Demeter-Guth and Wooley.
In the present paper, we introduce a multiple Ramanujan sum for arithmetic functions, which gives a multivariable extension of the generalized Ramanujan sum studied by D. R. Anderson and T. M. Apostol. We then find fundamental arithmetic properties of the multiple Ramanujan sum and study several types of Dirichlet series involving the multiple Ramanujan sum. As an application, we evaluate higher-dimensional determinants of higher-dimensional matrices, the entries of which are given by values of the multiple Ramanujan sum.
On higher dimensional arithmetic progressions in Meyer sets
本文研究Meyer集中高维等差数列的存在条件，证明当比率线性无关时，存在任意长度且具有k个线性无关比率的等差数列当且仅当k不超过相关完全欧几里得模型集生成的整数模的秩，并据此刻画了Meyer集作为完全欧几里得模型集子集的特征。
We give arithmetic formulas for the coefficients of Hauptmoduln of higher levels as analogues of Kaneko's result. We also obtain their asymptotic formulas by employing Murty-Sampath's method.
On the higher algebraic $K$-groups of arithmetically equivalent number fields
本文基于数域整数环的高阶代数K-群结构，引入K-等价与几乎K-等价，结合Rost-Voevodsky定理等，在p≠2时改进Komatsu的结果，明确相关等价条件及算术信息的共鸣关系。
We introduce geometric and homological finiteness properties for countable approximate groups via coarse geometry and then study these finiteness properties for S-arithmetic reductive approximate groups. For S-arithmetic approximate groups without infinite places we show that the finiteness length is finite and compute this finiteness length explicitly. In the simple case it is one less than the sum of the local ranks. This extends the Rank Theorem of Bux, Köhl and the second author from positive characteristic to characteristic zero. Our proof is based on a geometric version of their proof, b…
Conjugacy width in uniform higher rank arithmetic groups of orthogonal type
研究正交型各向异性高阶算术群的共轭宽度，在GRH下证明次数≥17且含非阿基米德赋值时宽度有界，并建立宽度界与同余子群问题的等价联系。
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