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.
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 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…
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.
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…
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.
In this article, given a scheme $X$ we show the existence of canonical lifts of Frobenius maps in an inverse system of schemes obtained from the fiber product of the canonical prolongation sequence of arithmetic jet spaces $J^*X$ and a prolongation sequence $S^*$ over the scheme $X$. As a consequence, for any smooth group scheme $E$, if $N^n$ denote the kernel of the canonical projection map of the $n$-th jet space $J^nE \rightarrow E$, then the inverse system $\{N^n\}_n$ is a prolongation sequence.
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.
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.
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.
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…
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 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 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.
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.
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 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…
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…
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.
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…
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.
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 (…
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…
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.
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 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 …
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.
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.
Let $k$ be a field of characteristic $0$. In this paper we describe a classification of smooth log K3 surfaces $X$ over $k$ whose geometric Picard group is trivial and which can be compactified into del Pezzo surfaces. We show that such an $X$ can always be compactified into a del Pezzo surface of degree $5$, with a compactifying divisor $D$ being a cycle of five $(-1)$-curves, and that $X$ is completely determined by the action of the absolute Galois group of $k$ on the dual graph of $D$. When $k=\mathbb{Q}$ and the Galois action is trivial, we prove that for any integral model $\mathcal{X}/\…
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…
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.
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.
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.
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.
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…
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…
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…
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…
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…
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…
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.
We give a new proof the arithmetic Hilbert-Samuel theorem by using classical reductions in the theory of coherent sheaves, a direct proof in the case of the projective space and the conservation of some numerical invariants, called arithmetic Hilbert invariants, through the deformation to the projective completion of the cone. This construction lies at the intersection of deformation theory and Arakelov geometry. It provides a deformation of a Hermitian line bundle over the deformation to the normal cone.
In this paper we shall study the inverse problem relative to dynamics of the w function which is a special arithmetic function and shall get some results.
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.
This is a textbook on arithmetic geometry with special regard to unramified Brauer groups of algebraic varieties. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, arithmetic and geometry of quadrics, Weil restriction of scalars, algebraic tori, an example of a stably rational non-rational variety, Brauer-Manin obstruction. All material is split into locally trivial problems with detailed hints.
Studying and comparing arithmetic properties of a given automatic sequence and the sequence of coefficients of the composition inverse of the associated formal power series (the formal inverse of that sequence) is an interesting problem. This problem was studied before for the Thue-Morse sequence. In this paper, we study arithmetic properties of the formal inverses of two sequences closely related to the well-known Baum-Sweet sequence. We give the recurrence relations for their formal inverses and we determine whether the sequences of indices at which these formal inverses take value $0$ and $…
