Huayi Chen, Atsushi Moriwaki
In this article, we consider an analogue of Arakelov theory of arithmetic surfaces over a trivially valued field. In particular, we establish an arithmetic Hilbert-Samuel theorem and studies the effectivity up to R-linear equivalence of pseudoeffective metrised R-divisors.
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The purpose of this paper is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with noncommutative arithmetic surfaces. We introduce a version of arithmetic intersection theory on noncommutative arithmetic surfaces and we prove an arithmetic Riemann-Roch theorem in this setup.
We study equivariant Arakelov-Euler characteristics of hermitian sheaves on arithmetic varieties which support a tame action by a finite group G. The tameness of the group action allows us to produce an equivariant Arakelov-Euler characteristic in a particularly fine "projective" arithmetic class group. We then show that the equivariant Arakelov-Euler characteristics of various complexes of differentials determine the epsilon constants of the L-functions of the motives obtained from the arithmetic variety using symplectic representations of the group G. Our results may be viewed firstly as a h…
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.
The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions.
We give an Arakelov theoretic proof of the equality of conductor and discriminant for arithmetic surfaces over number fields. This was first proved by T. Saito for relative curves over discrete valuation rings.
We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from $BGL$ to the Deligne cohomology spectrum. Secondly, Arakelov motivic cohomology is a generalization of arithmetic $K$-theory and arithmetic Chow groups. Finally, we give a conceptual explanation of the height pairing: it is given by the natural pairing of motivic homology and Arakelov motivic cohomology.
We study locally free sheaves of rank two on the projective line over the integers, especially indecomposable ones. Subsequently we apply various concepts of Arakelov geometry to these sheaves. We compute for example the arithmetic Chern classes and use the arithmetic Riemann-Roch theorem.
By some result on the study of arithemtic over trivially valued field, we find its applications to Arakelov geometry over adelic curves. We prove a partial result of the continuity of arithmetic $χ$-volume along semiample divisors. Moreover, we give a upper bound estimate of arithmetic Hilbert-Samuel function.
In the early 2000's Levine and Morel have given a geometric construction of an algebraic cobordism group defined for all smooth quasi projective varieties over a field. We show how we can refine their construction to build an Arakelov version of this group for Arakelov varieties over a number field, and how this integrates well in the general philosophy of Arakelov geometry.
This note contains an elementary discussion of the Arakelov intersection theory of elliptic curves. The main new results are a projection formula for elliptic arithmetic surfaces and a formula for the "energy" of an isogeny between Riemann surfaces of genus 1. The latter formula provides an answer to a question originally posed by Szpiro.
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…
In this paper we prove a concentration theorem for arithmetic $K_0$-theory, this theorem can be viewed as an analog of R. Thomason's result in the arithmetic case. We will use this arithmetic concentration theorem to prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. Such a formula was conjectured of a slightly stronger form by K. Köhler and D. Roessler and they also gave a correct route of its proof. Nevertheless our new proof is much simpler since it looks more natural and it doesn't involve too many complicated computations.
This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta functions formulated by the second author. Taking advantage of the six functors formalism in motivic stable homotopy theory, we establish a number of formal properties, including pullbacks for arbitrary morphisms, pushforwards for projective morphisms between regular schemes, localization sequences, $h$-descent. We round off the picture with a purity result an…
The $θ$-density in Arakelov geometry
本文为射影算术簇上丰富Hermitian线丛构造了$θ$-密度，证明其行为与Arakelov几何中仅考虑范数小于1的整体截面的通常密度相似，并由此证明了关于不可约性和正则性的两个Bertini型定理的类似结论。
In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. …
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 introduce the p-adic analogue of Arakelov intersection theory on arithmetic surfaces. The intersection pairing in an extension of the p-adic height pairing for divisors of degree 0 in the form described by Coleman and Gross. It also uses Coleman integration and is related to work of Colmez on p-adic Green functions. We introduce the p-adic version of a metrized line bundle and define the metric on the determinant of its cohomology in the style of Faltings. It is possible to prove in this theory analogues of the Adjunction formula and the Riemann-Roch formula.
This paper deals with Arakelov vector bundles over an arithmetic curve, i.e. over the set of places of a number field. The main result is that for each semistable bundle E, there is a bundle F such that $E \otimes F$ has at least a certain slope, but no global sections. It is motivated by an analogous theorem of Faltings for vector bundles over algebraic curves and contains the Minkowski-Hlawka theorem on sphere packings as a special case. The proof uses an adelic version of Siegel's mean value formula.
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 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 is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "`a la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut-Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.
We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of orthogonal flag varieties. We use these polynomials to describe the arithmetic Schubert calculus on these spaces. We also give a method to compute the natural arithmetic intersection numbers which arise, and prove that they are all rational numbers.
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 introduce the notion of an effective Arakelov divisor for a number field and the arithmetical analogue of the dimension of the space of sections of a line bundle. We study the analogue of the theta divisor for a number field.
Let X be the flag variety of the symplectic group. We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of X. We use these polynomials to describe the arithmetic Schubert calculus on X. Moreover, we give a method to compute the natural arithmetic Chern numbers on X, and show that they are all rational numbers.
We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution.
Arakelov geometry of Cuntz-Pimsner algebras
利用C*-代数的K理论研究Arakelov几何，证明V的Picard群与关联的Cuntz-Pimsner代数的K0群同构，并应用于数域上代数簇的有限性问题。
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.
By using the $\mathbb R$-filtration approach of Arakelov geometry, one establishes explicit upper bounds for geometric and arithmetic Hilbert-Samuel function for line bundles on projective varieties and hermitian line bundles on arithmetic projective varieties.
We prove an arithmetic analogue of Fujita's approximation theorem in Arakelov geometry, conjectured by Moriwaki, by using slope method and measures associated to $\mathbb R$-filtrations.
Arakelov geometry of toric bundles: Okounkov bodies and BKK
本文以 adelic 纤维丛观点研究环面丛，分析 Okounkov 体与 Boucksom-Chen 变换，并证明相交数公式的算术类比，用于计算半 Abel 簇环面紧化的高度和逐次最小值。
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 prove an analogue in Arakelov geometry of the Grothendieck-Riemann-Roch theorem.
Given a choice of metric on the Riemann surface, the regularized determinant of Laplacian (analytic torsion) is defined via the complex power of elliptic operators: $$ \det(Δ)=\exp(-ζ'(0)) $$ In this paper we gave an asymptotic effective estimate of analytic torsion under Arakelov metric. In particular, after taking the logarithm it is asymptotically upper bounded by $g$ for $g>1$. The construction of a cohomology theory for arithmetic surfaces in Arakelov theory has long been an open problem. In particular, it is not known if $h^{1}(X,L)\ge 0$. We view this as an indirect piece of evidence th…
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…
For an arithmetic surface $X\to B=\operatorname{Spec} O_K$ the Deligne pairing $\left <\,,\,\right > \colon \operatorname{Pic}(X) \times \operatorname{Pic}(X) \to \operatorname{Pic}(B)$ gives the "schematic contribution" to the Arakelov intersection number. We present an idelic and adelic interpretation of the Deligne pairing; this is the first crucial step for a full idelic and adelic interpretation of the Arakelov intersection number. For the idelic approach we show that the Deligne pairing can be lifted to a pairing $\left<\,,\,\right>_i:\ker(d^1_\times)\times \ker(d^1_\times)\to\operatorna…
We introduce an adelic Cartier divisor over a trivially valued field and discuss the bigness of it. For bigness, we give the integral representation of the arithmetic volume and prove the existence of limit of it. Moreover, we show that the arithmetic volume is continuous and log concave.
On a given arithmetic surface, inspired by work of Miyaoka, we consider vector bundles which are extensions of a line bundle by another one. We give sufficient conditions for their restriction to the generic fiber to be semi-stable. We then apply the arithmetic analog of Bogomolov inequality in Arakelov theory, and deduce from it a lower bound for some successive minima in the lattice of extension classes between these line bundles.
Uniform lower bound of arithmetic Hilbert--Samuel function of hypersurfaces
本文给出了射影超曲面算术Hilbert-Samuel函数具有最优主导项的显式一致下界，并将其应用于行列式方法中。
In this paper a concrete definition of higher K-theory in Arakelov geometry is given. The K-thoery defined in this paper is a higher extension of the arithmetic K_0 group of an arithmetic variety defined by Gillet and Soule. Products and direct images in this K-theory are discussed.
Riemann-Hurwitz Formula for Arithmetic Surfaces
本文基于算术曲面的正则性提出映射分解方法，并推导了算术曲面上的黎曼-Hurwitz公式，扩展至Arakelov理论。
In this article, we consider singular equivariant arithmetic schemes whose generic fibres are smooth. For such schemes, we prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. This formula is an analog, in the arithmetic case, of the Lefschetz formula proved by R. W. Thomason. In particular, our result implies a fixed point formula which was conjectured by V. Maillot and D. Rössler.
We give a new construction of higher arithmetic Chow groups for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analogue, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. The degree zero group agrees with the arithmetic Chow groups of Burgos. Our new construction is shown to be a contravariant functor and is endowed with a product structure, which is commutative and associative.
Let K be an algebraic number field and O_K the ring of integers of K. Let f : X --> Spec(O_K) be a stable arithmetic surface over O_K of genus g >= 2. In this short note, we will prove that if f has a reducible geometric fiber, then the self intersection of dualizing sheaf of X with Arakelov metric is greater than or equal to log(2)/6(g-1).
In the setting of Arakelov geometry over adelic curves, we introduce the $χ$-volume function and show some general properties. This article is dedicated to talk about the continuity of $χ$-volume function. By discussing its relationship with volume function, we prove its continuity around adelic $\mathbb{Q}$-ample $\mathbb{Q}$-Cartier divisors and its continuity in the trivially valued case. The study of the variation of arithmetic Okounkov bodies leads us to its continuous extension on arithmetic surfaces.
Arithmetic invariants of Euclidean lattice
本文研究了欧几里得格式的算术不变量，探讨了其在阿雷科洛夫几何中的应用，包括算术类比、有限性定理及梅林变换的积分表示。
In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element -- the arithmetic Atiyah class -- in a suitable arithmetic extension group. If $\overline{E}$ is a hermitian vector bundle on an arithmetic scheme $X$, its arithmetic Atiyah class is an obstruction to the algebraicity of the unitary connection on the vector bundle $E_\C$ over the complex manifold $X(\C)$ that is compatible with its holomorphic structure. We develop basic properties of the ar…
We consider heights of horizontal irreducible divisors on an arithmetic surface with respect to some hermitian line bundle. We obtain both lower and upper bounds for these heights. The results are different and sometimes stronger that those of S.Zhang on the same question. The case of the relative dualizing sheaf with the Arakelov metric is made especially explicit.
We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. We show that these notions can be translated in terms of convex analysis, and are closely related to objects like polyhedral complexes, concave functions, real Monge-Ampère measures, and Legendre-Fen…
We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and Pólya-Bertrandias valid over the projective line to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and $p$-adic curves). We also discuss geometric analogues, pe…
Let F be the complete flag variety over Spec(Z) with the tautological filtration 0 \subset E_1 \subset E_2 \subset ... \subset E_n=E of the trivial bundle E over F. The trivial hermitian metric on E(\C) induces metrics on the quotient line bundles L_i(\C). Let \hat{c}_1(L_i) be the first Chern class of L_i in the arithmetic Chow ring \hat{CH}(F) and x_i = -\hat{c}_1(L_i). Let h(X_1,...,X_n) be a polynomial with integral coefficients in the ideal <e_1,...,e_n> generated by the elementary symmetric polynomials e_i. We give an effective algorithm for computing the arithmetic intersection h(x_1,..…
This paper concerns complex algebraic K3 surfaces with an automorphism which acts trivially on the Neron-Severi group. Complementing a result by Vorontsov and Kondo, we determine those K3 surfaces where the order of the automorphism is a 2-power and equals the rank of the transcendental lattice. We also study the arithmetic of these K3 surfaces and comment on mirror symmetry
p-adic adelic metrics and Quadratic Chabauty I
本文提出了一种基于p-进阿雷克洛夫理论的新方法，用于构造abelian簇上的p-进高度，并简化了二次查巴乌方法用于计算有理点。
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