C. Deninger
This report on the topics in the title was written for a lecture series at the Southwestern Center for Arithmetic Algebraic Geometry at the University of Arizona.It may serve as an introduction to certain conjectural relations between number theory and the theory of dynamical systems on foliated spaces. The material is based on streamlined and updated versions of earlier papers on this subject.
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We point out analogies between (a) the explicit formulas in analytic number theory and transversal index theory, (b) Lichtenbaum's recent conjectures on special values of Hasse-Weil zeta functions and a formula for special values of Ruelle zeta functions, (c) work of Cramer on the zeroes of the Riemann zeta function and a result of Chazarain on the trace of a wave operator. These analogies suggest certain problems in the analysis on foliated spaces. The paper is mostly a review of previous work but the ideas concerning (c) are new.
A Riemmanian foliated dynamical system of 3-dimension $(\mathrm{RFDS}^{3})$ is a closed Riemannian 3-manifold with additional structures: foliation, dynamical system. In the context of arithmetic topology, it is a geometric/analytic analogue of an arithmetic scheme with a conjectural dynamical system suggested by C. Deninger. In this paper, we show leafwise cohomological expression of dynamical zeta function on a Riemannian foliated dynamical system.
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。
This paper is motivated by Deninger's programme. First we prove, using Alvarez Lopez-Kordyukov results, an Atiyah-Bott-Lefschetz trace formula for the cohomology groups associated to a ramified leafwise flat line bundle on a riemannian foliation. Then we argue, by precise computations, that cohomology groups associated to ramified leafwise flat vector bundles on a suitable foliated space (whose existence is still unknown) might be useful to study arithmetic L-functions via Atiyah-Bott-Lefschetz trace formulas.
We prove higher moment formulas for Siegel transforms defined over the space of unimodular $S$-lattices in $\mathbb Q_S^d$, $d\ge 3$, where in the real case, the formulas are introduced by Rogers (1955). As applications, we obtain the random statements of Gauss circle problem for any convex sets in $\mathbb Q_S^d$ containing the origin and of the effective Oppenheim conjecture for $S$-arithmetic quadratic forms.
Following recent work of the author, partly in collaboration with T. Dupuy and M. Barrett, we describe arithmetic analogues of some key concepts from Riemannian geometry such as: metrics, Chern connections, curvature, etc. Theorems are stated to the effect that the spectrum of the integers has a non-vanishing curvature.
This is the second paper in a series devoted to developing an arithmetic PDE analogue of Riemannian geometry. In Part 1 arithmetic PDE analogues of Levi-Civita and Chern connections were introduced and studied. In this paper arithmetic analogues of curvature and characteristic classes are developed.
We review the following subjects: 1. Basic theory on algebraic curves and their moduli space, 2. Schottky uniformization theory of Riemann surfaces, and its extension called arithmetic uniformization theory, 3. Application to these theories to the arithmetic of the moduli space of algebraic curves, especially to automorphic forms on this space.
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.
We define certain arithmetic derivatives on $\mathbb{Z}$ that respect the Leibniz rule, are additive for a chosen equation $a+b=c$, and satisfy a suitable non-degeneracy condition. Using Geometry of Numbers, we unconditionally show their existence with controlled size. We prove that any power-saving improvement on our size bounds would give a version of the $abc$ Conjecture. In fact, we show that the existence of sufficiently small arithmetic derivatives in our sense is equivalent to the $abc$ Conjecture. Our results give an explicit manifestation of an analogy suggested by Vojta in the eighti…
This is the first in a series on papers developing an arithmetic PDE analogue of Riemannian geometry. The role of partial derivatives is played by Fermat quotient operations with respect to several Frobenius elements in the absolute Galois group of a $p$-adic field. Existence and uniqueness of geodesics and of Levi-Civita and Chern connections are proved in this context. In a sequel to this paper a theory of arithmetic Riemannian curvature and characteristic classes will be developed.
We give a language for geometry which makes curves and number fields look alike.
We illustrate the principle: rational generating series occuring in arithmetic geometry are motivic in nature.
The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space.
The paper investigates some aspects of the geometry and the arithmetic of a non-rigid Calabi-Yau threefold. Particular emphasis is given to the study of its L-function L(H^3,s) and the Galois representation.
We describe connections between the Fourier coefficients of derivatives of Eisenstein series and invariants from the arithmetic geometry of the Shimura varieties $M$ associated to rational quadratic forms $(V,Q)$ of signature $(n,2)$. In the case $n=1$, we define generating series $\hatϕ_1(τ)$ for 1-cycles (resp. $\hatϕ_2(τ)$ for 0-cycles) on the arithmetic surface $\Cal M$ associated to a Shimura curve over $\Bbb Q$. These series are related to the second term in the Laurent expansion of an Eisenstein series of weight $\frac32$ and genus 1 (resp. genus 2) at the Siegel--Weil point, and these …
We study arithmetic distribution relations and the inverse function theorem in algebraic and arithmetic geometry, with an emphasis on versions that can be applied uniformly across families of varieties and maps. In particular, we prove two explicit versions of the inverse function theorem, the first via general distribution and separation inequalities that may be of independent interest, the second via a careful implementation of classical Newton iteration.
We define and investigate extension groups in the context of Arakelov geometry. The 'arithmetic extension groups' we introduce are extensions by groups of analytic types of the usual extension groups attached to $Ø_X$-modules over an arithmetic scheme $X$. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over $X$ - and we especially consider the case when $X$ is an 'arithmetic curve', namely the spectrum $\Spec Ø_K$ of the ring of integers in some number field $K$. T…
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。
Rigid, hard and soft problems and results in arithmetic geometry are presented. "Soft" and "hard" in our paper are limited to the framework of solutions of quadratic forms over rings of integers of local and global fields, the Hardy-Littlewood-Kloosterman method. Next we consider the notion of rigidity. In the framework we give review of some novel results in the aria.
Arithmetic Functions and Geometry
本文在统一的代数框架下重新审视Euler函数、除数和函数与Dedekind的$ψ$-函数，揭示其与几何的联系，并为未来类似构造提供背景，同时对Dedekind的$ψ$-函数提出了原创性观察。
Some Connections Between The Arithmetic and The Geometry of Lipschitz Integers
本文揭示了Lipschitz整数算术与几何间的联系，证明特定向量积的倍数性质，为Gordon Pall旧结果提供新算术证明，并提出与因式分解相关的几何问题。
This is a brief exposition of the mathematical themes that motivate the special programme at the Newton Institute in 2009. It is mostly intended for the general public having mathematical training up to the level of secondary school.
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.
We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$-th Gleason polynomial $G_n\in\mathbb{Q}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\mathrm{Per}_n(0)$ is the $1$-dimensional moduli space of quadratic rational maps on $\mathbb{P}^1$ with an $n$-periodic critical point. We show that if $G_n$ is irreducible over $\mathbb{Q}$, then $\mathrm{Per}_n(0)$ is irreducible over $\mathbb{C}$. To do this, we exhibit a $\mathbb{Q}$-rational smooth point on a projective completion of…
This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve. Formal-analytic surfaces provide a natural framework for arithmetic algebraization theorems, old and new. Formal-analytic arithmetic surfaces admit a rich geometry which parallels the geometry of complex analytic surfaces. Notably the dichotomy between pseudoconvexity and pseudoconcavity plays a central role in their geometry. Our study of formal-analytic arithmetic surf…
This paper is part of a series of papers where an arithmetic analogue of classical differential geometry is being developed. In this arithmetic differential geometry functions are replaced by integer numbers, derivations are replaced by Fermat quotient operators, and connections (respectively curvature) are replaced by certain adelic (respectively global) objects attached to symmetric matrices with integral coefficients. Previous papers were devoted to an arithmetic analogue of the Chern connection. The present paper is devoted to an arithmetic analogue of the Levi-Civita connection.
The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$ in $\mathbb{P}^r$. By means of a study of the finite dimensional vector spaces of global sections, we reduce the problem of constructing and finding efficiently computable polynomial maps defining the addition morphism or isogenies to linear algebra. We demonstrate the effectiveness of the method by improving the best known complexity for doubling and tripli…
Those notes rest on the Samuel Eilenberg Lectures I gave at Columbia University, NY, in the fall 2022. I thank all the mathematicians who participated in their elaboration, directly or indirectly. They are meant to be published as a Springer Lecture Notes. This arXiv version is not identical with the final version.
This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.
Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describ…
We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
Arithmetic Geometry of Non-Commutative Spaces with Large Centres
本文利用非交换形式形变理论引入多项式恒等代数的算术几何，获取交换角度无法观测的局部信息，为非交换框架下的算术几何深入研究奠基。
Geometry and arithmetic of semi-arithmetic Fuchsian groups
本文引入了半算术Fuchsian群的新几何不变量stretch，基于Riemannian质心概念，并利用算术Margulis引理证明了在算术维数、stretch和余面积有界时仅有有限多个共轭类，无界时则存在无限序列。
This is an introduction to the topics of the title, from the 2017 Grenoble Summer school on Arakelov geometry and arithmetic applications. We review Arithmetic intersection numbers, explain the definition of the height of a variety and its properties, both in the framework of classical Arakelov geometry and of Zhang's adelic formalism. We then discuss arithmetic ampleness and its application to the equidistribution theorem of Szpiro-Ullmo-Zhang-Yuan, both at complex and nonarchimedean places. We conclude with Ullmo-Zhang's proof of the Bogomolov conjecture over number fields.
In recent work with Bhatt and Morrow, we defined a new integral p-adic cohomology theory interpolating between etale and de Rham cohomology. An unexpected feature of this cohomology is that in coordinates, it can be computed by a q-deformation of the de Rham complex, which is thus canonical, at least in the derived category. In this short survey, we try to explain what we know about this phenomenon, and what can be conjectured to hold.
We give an overview of several of the mathematical works of Gilles Lachaud and provide a historical context. This is interspersed with some personal anecdotes highlighting many facets of his personality.
We reformulate Lehmer's question from 1933 and a question due to Schinzel and Zassenhaus from 1965 in terms of a comparison of the Mahler measures and the houses, respectively, of monic integer reciprocal and skew-reciprocal polynomials of the same degree. This entails that understanding the difference between orientation-preserving and orientation-reversing mapping classes is at least as complicated as answering these questions.
Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles appears to be closely related to an effective version of Faltings's theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of {\em Galois representations}, the structures linking motives to automorphic forms according to the Langl…
The Absolute Anabelian Geometry of Virtual Curves Arising from Sections of Arithmetic Fundamental Groups of Configuration Spaces
本文研究由配置空间算术基本群截面产生的带点虚拟曲线，证明其基本群论拉回在多个方面具有类似于域上曲线基本群的远阿贝尔性质。
We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity' maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced. Self-similar objects provide a framework in which, one can unite some results and conjectures in different mathematical frameworks. In some general situations, one can define a well-behaved notion of dimension for self-similar objects. Morphisms between self-similar objects are also defined and a categorical treatment of this concept is provided. We conclude by some ph…
Rudin conjectured that there are never more than c N^(1/2) squares in an arithmetic progression of length N. Motivated by this surprisingly difficult problem we formulate more than twenty conjectures in harmonic analysis, analytic number theory, arithmetic geometry, discrete geometry and additive combinatorics (some old and some new) which each, if true, would shed light on Rudin's conjecture.
We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of conformally-inequivalent such with all radii being reciprocals of integers. We then prove a result in the opposite direction: the "superintegral" ones exist only in finitely many "commensurability classes," all in dimensions below 30.
The Siegel-Weil formula in geometry and arithmetic
本文从经典、几何和算术三个视角审视西格尔 - 韦伊公式，回顾正定格椭圆theta级数的公式，讨论任意符号格高亏格情形，给出几何西格尔 - 韦伊公式，还提及（部分推测性的）算术西格尔 - 韦伊公式，揭示特殊圈次数与艾森斯坦级数中心导数的联系。
This note completes a talk given at the conference Curves over Finite Fields: past, present and future celebrating the publication the book {\em Rational Points on Curves over Finite Fields by J.-P. Serre and organised at Centro de ciencias de Benasque in june 2021. It discusses a part of the history of algebraic geometry codes together with some of their recent applications. A particular focus is done on the "multiplicative" structure of these codes, i.e. their behaviour with respect to the component wise product. Some open questions are raised and discussed.
The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more re…
The geometry and arithmetic of bielliptic Picard curves
本文研究双椭圆 Picard 曲线及其 Prym 阿贝尔曲面，证明 Torelli 定理并给出 QM 的几何证明，进而刻画 Galois 作用并分类有理 Mordell-Weil 挠子群。
Arithmetic and geometry of Markov polynomials
本文研究马尔可夫多项式系数在牛顿多边形上的算术与几何性质，提出若干新猜想，并在斐波那契数和佩尔数对应的特殊情形下给出证明。
Given the ensemble of random Gaussian Laplace eigenfunctions on the three-dimensional torus (`3d arithmetic random waves'), we investigate the $1$-dimensional Hausdorff measure of the nodal intersection curve against a compact regular toral surface (the `nodal intersection length'). The expected length is proportional to the square root of the eigenvalue, times the surface area, independent of the geometry. Our main finding is the leading asymptotic of the nodal intersection length variance, against a surface of nonvanishing Gauss-Kronecker curvature. The problem is closely related to the theo…
We introduce the positive intersection product in Arakelov geometry and prove that the arithmetic volume function is continuously differentiable. As applications, we compute the distribution function of the asymptotic measure of a Hermitian line bundle and several other arithmetic invariants.
We study a K3 surface, which appears in the two-loop mixed electroweak-quantum chromodynamic virtual corrections to Drell--Yan scattering. A detailed analysis of the geometric Picard lattice is presented, computing its rank and discriminant in two independent ways: first using explicit divisors on the surface and then using an explicit elliptic fibration. We also study in detail the elliptic fibrations of the surface and use them to provide an explicit Shioda--Inose structure. Moreover, we point out the physical relevance of our results.
In this paper, we study the monodromy of Appell hypergeometric partial differential equations, which lead us to find four derivatives which are associated to the group GL(3). Our four derivatives have the remarkable properties. We find that Appell hypergeometric partial differential equations can be reduced to four nonlinear partial differential equations by the use of our four derivatives. We investigate these four nonlinear partial differential equations. We are interested in some particular cases. In these cases, the system of linear partial differential equations has the remarkable propert…
In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic forms), rigid Calabi-Yau threefolds and their arithmetic (Diophantine geometry) are intimately related to Hessian polyhedra and their invariants. We construct a Galois representation whose image is a proper subgroup of $W(E_6)$, the Weyl group of the exceptional Lie algebra $E_6$. We give a conjecture about the identification of two different kinds of $L$-function…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned with the problem of bounding the behaviour of arbitrary finite collections of geometric objects such as points, lines, or circles with respect to geometric operations such as incidence or distance. Given the presence of arbitrary finite sets in these problems, the methods used to attack these problems have primarily been combinatorial in na…
This article describes results of joint work with Michael Rapoport and Tonghai Yang. First, we construct an modular form ϕ(τ) of weight 3/2 valued in the arithmetic Chow group of the arithmetic surface M attached toa Shimura curve over Q. The q-expansion of this function is an analogue of the Hirzebruch-Zagier generating function for the cohomology classes of curves on a Hilbert modular surface. This`arithmetic theta function' is used to define an `arithmetic theta lift' from modular forms of weight 3/2 to the arithmetic Chow group of M. For integers t_1 and t_2 with t_1t_2 not a square, the (…
Existential closedness of $\overline{\mathbb{Q}}$ as a globally valued field via Arakelov geometry
本研究利用算术体积函数可微性与算术Bertini型定理分类闭点高度逼近问题，证明了$\overline{\mathbb{Q}}$作为全局赋值域（GVF）的存在封闭性，引入GVF泛函并给出Zhang不等式等号成立的新判据。
We study the geometry and arithmetic of so-called primary Burniat surfaces, a family of surfaces of general type arising as smooth bidouble covers of a del Pezzo surface of degree 6 and at the same time as étale quotients of certain hypersurfaces in a product of three elliptic curves. We give a new explicit description of their moduli space and determine their possible automorphism groups. We also give an explicit description of the set of curves of geometric genus 1 on each primary Burniat surface. We then describe how one can try to obtain a description of the set of rational points on a giv…
In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the ``closed fibers at infinity''. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus $g\geq 2$. We use Connes' theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger's Archimedean cohomology…
Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry and arithmetic of curves Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's and Campana's conjectures. Lecture 2: Campana's program; Campana constellations framed in terms of b-divisors, to allow for a definition of Kodaira dimension directly on the base. A speculative notion of firmaments which may deserve further investigation, espec…
We present an Arakelov theoretic version of the deformation to the normal cone. In particular, the geometric data is enriched with a deformation of a Hermitian line bundle. We introduce numerical invariants called arithmetic Hilbert invariants and prove the conservation of these invariants along the deformation. In a following article, this conservation of number theorem will allow a demonstration of the arithmetic Hilbert-Samuel theorem.
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