For instance, when 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 , give rise to arithmetic intersection numbers, which are real numbers when their geometric counterparts are integers.
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We might therefore expect that the more modern and more sophisticated tools of algebraic geometry which is a subject that started out as just the geometry of shapes described by polynomial equations might be extremely useful in answering questions and problems in number theory.
An example of this is the scheme , which is two-dimensional, and hence also referred to as an arithmetic surface. We recall that the points of an affine scheme , for some ring , are given by the prime ideals of.
It is important to consider the infinite primes when dealing with questions and problems in number theory, and therefore we need to modify some aspects of algebraic geometry if we are to use it in helping us with number theory.
We now go back to arithmetic schemes, taking into consideration the infinite prime. Since we are dealing with the ordinary integers , there is only one infinite prime, corresponding to the embedding of the rational numbers into the real numbers. More generally we can also consider an arithmetic variety over instead of is the ring of integers of a number field. In this case we may have several infinite primes, corresponding to the embediings of into the real and complex numbers.
In this post, however, we will consider only and one infinite prime. Let us look at the fibers of the arithmetic scheme. The fiber of an arithmetic scheme at a finite prime is given by the scheme defined by the same homogeneous polynomials as , but with the coefficients taken modulo , so that they are elements of the finite field.
The fiber over the generic point is given by taking the tensor product of the coordinate ring of with the rational numbers. But how should we describe the fiber over the infinite prime?
It was the idea of the mathematician Suren Arakelov that the fiber over the infinite prime should be given by a complex variety — in the case of an arithmetic surface, which Arakelov originally considered, the fiber should be given by a Riemann surface.
Arakelov unfortunately was not able to carry out his proof. He had only a very short career, being forced to retire after being diagnosed with schizophrenia. The space of differential forms see Differential Forms of degree on.
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Share this page Atsushi Moriwaki 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. Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes.