Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

* Provides an entry for graduate students into an active field of research
 * Each chapter includes exercises, examples, and figures
 * Will become a standard reference for researchers in the field
 * Contains descriptions of many known results and conjectures, together with an extensive bibliography

This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function.

A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs. As is typical in any subject combining Diophantine problems and geometry, a fundamental goal is to describe arithmetic properties, at least qualitatively, in terms of underlying geometric structures.

Key features:

- Provides an entry for graduate students into an active field of research

- Provides a standard reference source for researchers

- Includes numerous exercises and examples

- Contains a description of many known results and conjectures, as well as an extensive glossary, bibliography, and index

This graduate-level text assumes familiarity with basic algebraic number theory. Other topics, such as basic algebraic geometry, elliptic curves, nonarchimedean analysis, and the theory of Diophantine approximation, are introduced and referenced as needed. Mathematicians and graduate students will find this text to be an excellent reference.

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The Arithmetic of Dynamical Systems

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

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The Algebraic and Geometric Theory of Quadratic Forms

The first goal of algebraic number theory is the generalization of the theorem on the unique representation of natural numbers as products of prime numbers to algebraic numbers. Gauss considered the ring $ \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] $ of all numbers of the form $ a + \sqrt {{ - 1b}} $ with $ a,\;b \in \mathbb{Z} $ and showed that $ \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] $ is a ring with unique factorization in prime elements (see §2.1). He introduced these numbers for the development of his theory of biquadratic residues. Another motivation for the study of the arithmetic of algebraic numbers comes from the theory of Tiophantine equations. For example, the quadratic form $ f({x_1},\;{x_2}) = x_1^2 - Dx_2^2 $ with $ D \in \mathbb{Z} $, $ \sqrt {{D\; \notin \;\mathbb{Z}}} $ can be written in the form $ \left( {{x_1} - \sqrt {{D{x_2}}} } \right)\left( {{x_1} + \sqrt {{D{x_2}}} } \right) $. Hence the question about the representation of integers by f(a 1, a 2) with $ {a_1},\;{a_2} \in \mathbb{Z} $ can be reformulated as a question of factorization of algebraic numbers of the form $ {a_1} + \sqrt {{D{a_2}}} $. These numbers form a module in the field $ \mathbb{Q}(\sqrt {D} ) $.

Basic Number Theory

Kummer theory for abelian varieties over local fields

1: Complete discrete valuation fields 2: extensions of discrete valuation fields 3: the norm map 4: local class field theory I 5: local class field theory II 6: the group of units in a *rp-adic field 7: explicit formulas for the Hilbert norm residue symbold 8: explicit formulas for the Hilbert pairing on formal groups 9: the Milnor *K-groups of a local field.

Local fields and their extensions : a constructive approach

The functional equation of the twisted zeta function of algebraic number fields was first proved by Hecke (for a recent exposition see [N, Ch. VII]). Tate [T] and, for unramified characters without local theory, Iwasawa [I1–I2] lifted the zeta function to a zeta integral defined on an adelic space. Their method of proving the functional equation and deriving finiteness of several number theoretical objects is a generalization of one of the proofs of the functional equation by Riemann. The latter in the case of rational numbers uses an appropriate theta formula derived from a summation formula which itself follows from properties of Fourier transform. Hence the functional equation of the zeta integral reflects symmetries of the Fourier transform on the adelic object and its quotients, and the right mixture of their multiplicative and additive structures. With slight modification the approach of Tate and Iwasawa for characteristic zero can be extended to a uniform treatment of any characteristic, e.g. [W2]. Earlier, the functional field case was treated similarly to Hecke’s method by Witt in 1936 (cf. [Rq, sect. 7.4]), and it was also established using essentially harmonic analysis on finite rings by Schmid and Teichmüller [ST] in 1943 (for a modern exposition see e.g. [M,3.5]); those works were not widely known.

Analysis on arithmetic schemes. I.

For a higher local field F algebraic properties of Km(F )=\l>1lKm(F ) of the Milnor K -group are studied by using topological and arithmetical considerations. In particular, standartness of the torsion in the latter group and divisibility of the group \l>1lKm(F ) if the last residue field of F is finite are proved. It is shown that \l>1lKm(F ) coincides with the intersection of all neighbourhoods of zero in Km(F ) (and if F is an m-dimensional local field, it equals to the kernel of the reciprocity map Km(F )! Gal(F ab =F ) ). A description of Km(F )=l in the language of topological generators and relations is provided. The equality of several topologies on Km(F ) at the level of subgroups is shown and their difference is discussed. Appendix of this paper written by O. Izhboldin describes a construction of a field F containing a primitive p th root and such that p-torsion of Km(F )=\l>1lKm(F ) is not generated by p-torsion in F. He works with the field of rational functions of an infinite product of certain Severi-Brauer varieties.

Sequential topologies and quotients of Milnor K -groups of higher local fields

In this work we extend the first part of the previous paper [F4] to higher dimensional local fields. We introduce a nontrivial translation invariant measure on the additive group of higher dimensional local fields, and then develop elements of integration and harmonic analysis. We also discuss its relation with several other measure theories. For an n-dimensional local field F a translation invariant measure μ is defined on a certain ring A of measurable sets and takes values in R ((X1)) . . . ((Xn−1)) (which itself is an n-dimensional local field whose last residue field is the archimedean field R). The algebra A in the case of higher dimensional fields with finite last residue field is the algebra generated by characteristic functions of shifts of fractional ideals, i.e. a + bO with a, b ∈ F and O the ring of integers of F with respect to the n-dimensional structure. The measure is countably additive in a refined sense (see sections 7 and 8). Elements of integration theory are introduced in sections 9–13. The additive group of a higher dimensional field has a certain topology on it with respect to which it is not locally compact for n > 1. This topology plays a key role in higher class field theory [F1–F3]. The additive group of F is self dual, which together with the measure and integration leads to analogs of many classical results in Fourier analysis (section 14). In particular, for functions in certain space we introduce their transform

Measure, integration and elements of harmonic analysis on generalized loop spaces

Group theorists have recently been interested in a pro-p group known as the Nottingham group. This group $\mathcal{N}\left( {{\mathbb{F}_p}} \right)$is defined as the set of power series $t + {a_2}{t^2} + \ldots \in t\mathbb{F}p\left[ {\left[ t \right]} \right]$, the group operation being substitution of one power series in another. Interest in this group reached a peak with the proof by Rachel Camina of the following:

Where the Wild Things Are: Ramification Groups and the Nottingham Group

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