Around 1994 R. Borcherds discovered a new type of meromorphic modular form on the orthogonal group $O(2,n)$. These "Borcherds products" have infinite product expansions analogous to the Dedekind eta-function. They arise as multiplicative liftings of elliptic modular forms on $(SL)_2(R)$. The fact that the zeros and poles of Borcherds products are explicitly given in terms of Heegner divisors makes them interesting for geometric and arithmetic applications. In the present text the Borcherds' construction is extended to Maass wave forms and is used to study the Chern classes of Heegner divisors. A converse theorem for the lifting is proved.

Cohomology of Drinfeld Modular Varieties aims to provide an introduction to both the subject of the title and the Langlands correspondence for function fields. These varieties are the analogs for function fields of Shimura varieties over number fields. This present volume is devoted to the geometry of these varieties and to the local harmonic analysis needed to compute their cohomology. To keep the presentation as accessible as possible, the author considers the simpler case of function rather than number fields; nevertheless, many important features can still be illustrated. It will be welcomed by workers in number theory and representation theory.

The decomposition of the space L2(G(Q)\G(A)), where G is a reductive group defined over Q and A is the ring of adeles of Q, is a deep problem at the intersection of number and group theory. Langlands reduced this decomposition to that of the (smaller) spaces of cuspidal automorphic forms for certain subgroups of G. This book describes this proof in detail. The starting point is the theory of automorphic forms, which can also serve as a first step towards understanding the Arthur-Selberg trace formula. To make the book reasonably self-contained, the authors also provide essential background in subjects such as: automorphic forms; Eisenstein series; Eisenstein pseudo-series, and their properties. It is thus also an introduction, suitable for graduate students, to the theory of automorphic forms, the first written using contemporary terminology. It will be welcomed by number theorists, representation theorists and all whose work involves the Langlands program.

This volume has grown out of lectures given by Professor Pfister over many years. The emphasis here is placed on results about quadratic forms that give rise to interconnections between number theory, algebra, algebraic geometry and topology. Topics discussed include Hilbert's 17th problem, the Tsen-Lang theory of quasi algebraically closed fields, the level of topological spaces and systems of quadratic forms over arbitrary fields. Whenever possible proofs are short and elegant, and the author's aim was to make this book as self-contained as possible. This is a gem of a book bringing together thirty years' worth of results that are certain to interest anyone whose research touches on quadratic forms.

This book represents the refereed proceedings of the Fourth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing which was held at Hong Kong Baptist University in 2000. An important feature are invited surveys of the state-of-the-art in key areas such as multidimensional numerical integration, low-discrepancy point sets, random number generation, and applications of Monte Carlo and quasi-Monte Carlo methods. These proceedings include also carefully selected contributed papers on all aspects of Monte Carlo and quasi-Monte Carlo methods. The reader will be informed about current research in this very active field.

It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. O. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Oulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not' grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory arid the struc ture of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems," "chaos, synergetics and large-5cale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This program, Mathematics and Its Applications, is devoted to such (new) interrelations as exampla gratia: - a central concept which plays an important role in several different mathe matical and/or scientific specialized areas; - new applications of the results and ideas from one area of scientific en deavor into another; - influences which the results, problems and concepts of one field of enquiry have and have had on the development of another."

This book presents a broad, user-friendly introduction to the Langlands program, that is, the theory of automorphic forms and its connection with the theory of L-functions and other fields of mathematics. Each of the twelve chapters focuses on a particular topic devoted to special cases of the program. The book is suitable for graduate students and researchers.

This is the fourteenth annual volume arising from the Seminaire de Theorie des Nombres de Paris. As with previous volumes the whole spectrum of number theory is discussed, with many contributions from some of the world's leading figures. The very latest research developments are covered and much of the work presented here will not be found elsewhere. Also included are surveys that will serve to guide the reader through the extensive published literature. This will be a necessary addition to the libraries of all workers in number theory.

There is now a large body of theory concerning algebraic varieties over finite fields, and many conjectures exist in this area that are of great interest to researchers in number theory and algebraic geometry. This book is concerned with the arithmetic of diagonal hypersurfaces over finite fields, with special focus on the Tate conjecture and the Lichtenbaum-Milne formula for the central value of the L-function. It combines theoretical and numerical work, and includes tables of Picard numbers. Although this book is aimed at experts, the authors have included some background material to help non-specialists gain access to the results.

Dieses zweibAndige Werk handelt von Mathematik und ihrer Geschichte. Die sorgfAltige Analyse dessen, was die Alten bewiesen - meist sehr viel mehr, als sie ahnten -, fA1/4hrt zu einem besseren VerstAndnis der Geschichte und zu einer guten Motivation und einem ebenfalls besseren VerstAndnis heutiger Mathematik.
Die Themen des ersten Bandes reichen von der Konstruktion der reellen Zahlen mittels dedekindscher Schnitte bis hin zum Fundamentalsatz der Algebra. Dazwischen werden die BA1/4cher V bis X der euklidischen Elemente abgehandelt, wobei insbesondere die eudoxische Proportionenlehre (Buch V) eine zentrale Rolle spielt. Sie bietet einen eleganten Zugang zu den Logarithmen, so dass auch Neper ausfA1/4hrlich zu Wort kommt. Weitere Themen sind die natA1/4rlichen Zahlen und das Induktionsprinzip; die Entdeckung der LAsungsformeln der Gleichungen dritten und vierten Grades; Polynomringe in beliebig vielen Unbestimmten; symmetrische Polynome und der Satz von Waring.

This is a modern introduction to the analytic techniques used in the investigation of zeta-function. Riemann introduced this function in connection with his study of prime numbers, and from this has developed the subject of analytic number theory. Since then, many other classes of "zeta-function" have been introduced and they are now some of the most intensively studied objects in number theory. Professor Patterson has emphasized central ideas of broad application, avoiding technical results and the customary function-theoretic approach.

It is an historical goal of algebraic number theory to relate all algebraic extensionsofanumber?eldinauniquewaytostructuresthatareexclusively described in terms of the base ?eld. Suitable structures are the prime ideals of the ring of integers of the considered number ?eld. By examining the behaviouroftheprimeidealswhenembeddedintheextension?eld, su?cient information should be collected to distinguish the given extension from all other possible extension ?elds. The ring of integers O of an algebraic number ?eld k is a Dedekind ring. k Any non-zero ideal in O possesses therefore a decomposition into a product k of prime ideals in O which is unique up to permutations of the factors. This k decomposition generalizes the prime factor decomposition of numbers in Z Z. In order to keep the uniqueness of the factors, view has to be changed from elements of O to ideals of O . k k Given an extension K/k of algebraic number ?elds and a prime ideal p of O, the decomposition law of K/k describes the product decomposition of k the ideal generated by p in O and names its characteristic quantities, i. e. K the number of di?erent prime ideal factors, their respective inertial degrees, and their respective rami?cation indices. Whenlookingatdecompositionlaws, weshouldinitiallyrestrictourselves to Galois extensions. This special case already o?ers quite a few di?culties

Leibniz Algebras: Structure and Classification is designed to introduce the reader to the theory of Leibniz algebras. Leibniz algebra is the generalization of Lie algebras. These algebras preserve a unique property of Lie algebras that the right multiplication operators are derivations. They first appeared in papers of A.M Blokh in the 1960s, under the name D-algebras, emphasizing their close relationship with derivations. The theory of D-algebras did not get as thorough an examination as it deserved immediately after its introduction. Later, the same algebras were introduced in 1993 by Jean-Louis Loday , who called them Leibniz algebras due to the identity they satisfy. The main motivation for the introduction of Leibniz algebras was to study the periodicity phenomena in algebraic K-theory. Nowadays, the theory of Leibniz algebras is one of the more actively developing areas of modern algebra. Along with (co)homological, structural and classification results on Leibniz algebras, some papers with various applications of the Leibniz algebras also appear now. However, the focus of this book is mainly on the classification problems of Leibniz algebras. Particularly, the authors propose a method of classification of a subclass of Leibniz algebras based on algebraic invariants. The method is applicable in the Lie algebras case as well. Features: Provides a systematic exposition of the theory of Leibniz algebras and recent results on Leibniz algebras Suitable for final year bachelor's students, master's students and PhD students going into research in the structural theory of finite-dimensional algebras, particularly, Lie and Leibniz algebras Covers important and more general parts of the structural theory of Leibniz algebras that are not addressed in other texts

This book contains seventeen contributions made to George Andrews on the occasion of his sixtieth birthday, ranging from classical number theory (the theory of partitions) to classical and algebraic combinatorics. Most of the papers were read at the 42nd session of the Séminaire Lotharingien de Combinatoire that took place at Maratea, Basilicata, in August 1998.This volume contains a long memoir on Ramanujan's Unpublished Manuscript and the Tau functions studied with a contemporary eye, together with several papers dealing with the theory of partitions. There is also a description of a maple package to deal with general q-calculus. More subjects on algebraic combinatorics are developed, especially the theory of Kostka polynomials, the ice square model, the combinatorial theory of classical numbers, a new approach to determinant calculus.

From the reviews: "...a fine book ... treats algebraic number theory from the valuation-theoretic viewpoint. When it appeared in 1949 it was a pioneer. Now there are plenty of competing accounts. But Hasse has something extra to offer. This is not surprising, for it was he who inaugurated the local-global principle (universally called the Hasse principle). This doctrine asserts that one should first study a problem in algebraic number theory locally, that is, at the completion of a vaulation. Then ask for a miracle: that global validity is equivalent to local validity. Hasse proved that miracles do happen in his five beautiful papers on quadratic forms of 1923-1924. ... The exposition is discursive. ... It is trite but true: Every number-theorist should have this book on his or her shelf." (Irving Kaplansky in Bulletin of the American Mathematical Society, 1981)

Approach your problems from It isn't that they can't see the the right end and begin with the solution. It is that they can't see the problem. answers. Then, one day, perhaps you will find the final question. The Hermit Clad in Crane Feathers' G. K. Chesterton, The scandal of in R. Van Gulik's The Chinese Maze Father Brown "The point ofa pin" Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e DEGREES(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the s

The theme of this book is the characterization of certain multiplicative and additive arithmetical functions by combining methods from number theory with some simple ideas from functional and harmonic analysis. The authors achieve this goal by considering convolutions of arithmetical functions, elementary mean-value theorems, and properties of related multiplicative functions. They also prove the mean-value theorems of Wirsing and Halasz and study the pointwise convergence of the Ramanujan expansion. Finally, some applications to power series with multiplicative coefficients are included, along with exercises and an extensive bibliography.

Grothendieck's duality theory for coherent cohomology is a fundamental tool in algebraic geometry and number theory, in areas ranging from the moduli of curves to the arithmetic theory of modular forms. Presented is a systematic overview of the entire theory, including many basic definitions and a detailed study of duality on curves, dualizing sheaves, and Grothendieck's residue symbol. Along the way proofs are given of some widely used foundational results which are not proven in existing treatments of the subject, such as the general base change compatibility of the trace map for proper Cohen-Macaulay morphisms (e.g., semistable curves). This should be of interest to mathematicians who have some familiarity with Grothendieck's work and wish to understand the details of this theory.

This volume is devoted to the Brauer group of a commutative ring and related invariants. Part I presents a new self-contained exposition of the Brauer group of a commutative ring. Included is a systematic development of the theory of Grothendieck topologies and etale cohomology, and discussion of topics such as Gabber's theorem and the theory of Taylor's big Brauer group of algebras without a unit. Part II presents a systematic development of the Galois theory of Hopf algebras with special emphasis on the group of Galois objects of a cocommutative Hopf algebra. The development of the theory is carried out in such a way that the connection to the theory of the Brauer group in Part I is made clear. Recent developments are considered and examples are included.

The Brauer-Long group of a Hopf algebra over a commutative ring is discussed in Part III. This provides a link between the first two parts of the volume and is the first time this topic has been discussed in a monograph.

Audience: Researchers whose work involves group theory. The first two parts, in particular, can be recommended for supplementary, graduate course use. "

Many classical and modern results and quadratic forms are brought together in this book. The treatment is self-contained and of a totally elementary nature requiring only a basic knowledge of rings, fields, polynomials, and matrices, such that the works of Pfister, Hilbert, Hurwitz and others are easily accessible to non-experts and undergraduates alike. The author deals with many different approaches to the study of squares; from the classical works of the late 19th century, to areas of current research. Anyone with an interest in algebra or number theory will find this a most fascinating volume.

In this tract, Professor Moreno develops the theory of algebraic curves over finite fields, their zeta and L-functions, and, for the first time, the theory of algebraic geometric Goppa codes on algebraic curves. Among the applications considered are: the problem of counting the number of solutions of equations over finite fields; Bombieri's proof of the Reimann hypothesis for function fields, with consequences for the estimation of exponential sums in one variable; Goppa's theory of error-correcting codes constructed from linear systems on algebraic curves; there is also a new proof of the TsfasmanSHVladutSHZink theorem. The prerequisites needed to follow this book are few, and it can be used for graduate courses for mathematics students. Electrical engineers who need to understand the modern developments in the theory of error-correcting codes will also benefit from studying this work.

This is a new approach to the theory of non-holomorphic modular forms, based on ideas from quantization theory or pseudodifferential analysis. Extending the Rankin-Selberg method so as to apply it to the calculation of the Roelcke-Selberg decomposition of the product of two Eisenstein series, one lets Maass cusp-forms appear as residues of simple, Eisenstein-like, series. Other results, based on quantization theory, include a reinterpretation of the Lax-Phillips scattering theory for the automorphic wave equation, in terms of distributions on R2 automorphic with respect to the linear action of SL(2, Z)

Peter Giblin describes, in the context of an introduction to the theory of numbers, some of the more elementary methods for factorization and primality testing; that is, methods independent of a knowledge of other areas of mathematics. Indeed everything is developed from scratch so the mathematical prerequisites are minimal. An essential feature of the book is the large number of computer programs (written in Pascal) and a wealth of computational exercises and projects, in addition to more usual theory exercises. The theoretical development includes continued fractions and quadratic residues, directed always towards the two fundamental problems of primality testing and factorization. There is time, all the same, to include a number of topics and projects of a purely "recreational" nature.