Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the classical birational geometry, i. After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional varieties. The book includes such fundamental results as arithmetic Hilbert—Samuel formula, arithmetic Nakai—Moishezon criterion, arithmetic Bogomolov inequality, the existence of small sections, the continuity of arithmetic volume function, the Lang—Bogomolov conjecture and so on. In addition, the author presents, with full details, the proof of Faltings' Riemann—Roch theorem.

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A combination of the Grothendieck algebraic geometry of schemes over with Hermitian complex geometry on their set of complex points. The goal is to provide a geometric framework for the study of Diophantine problems in higher dimension cf.

The construction relies upon the analogy between number fields and function fields: the ring has Krull dimension cf. Dimension one, and "adding a point" to the corresponding scheme makes it look like a complete curve. For instance, if is a rational number, the identity. In higher dimension, given a regular projective flat scheme over , one considers pairs consisting of an algebraic cycle of codimension over , together with a Green current for on the complex manifold : is real current of type such that, if denotes the current given by integration on , the following equality of currents holds:.

Equivalence classes of such pairs form the arithmetic Chow group , which has good functoriality properties and is equipped with a graded intersection product, at least after tensoring it by.

These notions were first introduced for arithmetic surfaces, i. For the general theory, see [a7] , [a9] and references therein. Given a pair consisting of an algebraic vector bundle on and a Hermitian metric on the corresponding holomorphic vector bundle on the complex-analytic manifold , one can define characteristic classes of with values in the arithmetic Chow groups of. For instance, when has rank one, if is a non-zero rational section of and its divisor, the first Chern class of is the class of the pair.

The main result of the theory is the arithmetic Riemann—Roch theorem, which computes the behaviour of the Chern character under direct image [a8] , [a6]. Its strongest version involves regularized determinants of Laplace operators and the proof requires hard analytic work, due to J.

Bismut and others. Since , the pairings. Examples of such real numbers are the heights of points and subvarieties, for which Arakelov geometry provides a useful framework [a3].

When is a semi-stable arithmetic surface, an important invariant of is the self-intersection of the relative dualizing sheaf equipped with the Arakelov metric [a1]. Szpiro and A. Parshin have shown that a good upper bound for would lead to an effective version of the Mordell conjecture and to a solution of the ABC conjecture [a10]. Faltings and E. Vojta used Arakelov geometry to give a new proof of the Mordell conjecture [a12] , by adapting the method of Diophantine approximation.

More generally, Faltings obtained by Vojta's method a proof of a conjecture of S. Lang on Abelian varieties [a5] : Assume is an Abelian variety over a number field and let be a proper closed subvariety in ; then the set of rational points of is contained in the union of finitely many translates of Abelian proper subvarieties of.

See also Diophantine geometry ; Height, in Diophantine geometry ; Mordell conjecture. Log in. Namespaces Page Discussion. Views View View source History. Jump to: navigation , search. Arakelov theory A combination of the Grothendieck algebraic geometry of schemes over with Hermitian complex geometry on their set of complex points.

For instance, if is a rational number, the identity where is the valuation of at the prime and where , is similar to the Cauchy residue formula for the differential , when is a non-zero rational function on a smooth complex projective curve. In higher dimension, given a regular projective flat scheme over , one considers pairs consisting of an algebraic cycle of codimension over , together with a Green current for on the complex manifold : is real current of type such that, if denotes the current given by integration on , the following equality of currents holds: where is a smooth form of type.

Since , the pairings , give rise to arithmetic intersection numbers, which are real numbers when their geometric counterparts are integers.

References [a1] S. Arakelov, "Intersection theory of divisors on an arithmetic surface" Math. USSR Izv. Arakelov, "Theory of intersections on an arithmetic surface" , Proc. Mathematicians Vancouver , 1 , Amer. Bost, H. Gillet, C. Faltings, "Calculus on arithmetic surfaces" Ann. Faltings, "Diophantine approximation on Abelian varieties" Ann. Faltings, "Lectures on the arithmetic Riemann—Roch theorem" Ann.

Study , Notes by S. Zhang MR Zbl IHES , 72 pp. Abramovich, J. Burnol, J. Kramer, "Lectures on Arakelov geometry" , Studies Adv. Press MR Zbl Vojta, "Siegel's theorem in the compact case" Ann. How to Cite This Entry: Arakelov geometry.

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A combination of the Grothendieck algebraic geometry of schemes over with Hermitian complex geometry on their set of complex points. The goal is to provide a geometric framework for the study of Diophantine problems in higher dimension cf. The construction relies upon the analogy between number fields and function fields: the ring has Krull dimension cf. Dimension one, and "adding a point" to the corresponding scheme makes it look like a complete curve. For instance, if is a rational number, the identity.

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## Arakelov Geometry

It seems that you're in Germany. We have a dedicated site for Germany. Authors: Chen , Huayi, Moriwaki , Atushi. The purpose of this book is to build the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for research on arithmetic geometry in several directions. By adelic curve is meant a field equipped with a family of absolute values parametrized by a measure space, such that the logarithmic absolute value of each non-zero element of the field is an integrable function on the measure space. They then establish a theory of adelic vector bundles on adelic curves, which considerably generalizes the classic geometry of vector bundles or that of Hermitian vector bundles over an arithmetic curve.