Number theory Books

Book SynopsisThis book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms. Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.Trade Review“This is a very interesting monograph providing a fast introduction to different fields of research devoted to modern aspects and develompents of commutative algebra, algebraic geometry, combinatorics, etc.” (Piotr Pokora, zbMATH 1445.13001, 2020)Table of Contents- Part I Associated Primes of Powers of Ideals - Associated Primes of Powers of Ideals. - Associated Primes of Powers of Squarefree Monomial Ideals. - Final Comments and Further Reading. - Part II Regularity of Powers of Ideals. - Regularity of Powers of Ideals and the Combinatorial Framework. - Problems, Questions, and Inductive Techniques. - Examples of the Inductive Techniques. - Final Comments and Further Reading. - Part III The Containment Problem. - The Containment Problem: Background. - The Containment Problem. - The Waldschmidt Constant of Squarefree Monomial Ideals. - Symbolic Defect. - Final Comments and Further Reading. - Part IV Unexpected Hypersurfaces. - Unexpected Hypersurfaces. - Final Comments and Further Reading.

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Book SynopsisThis open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.Trade Review“The book contains a huge amount of interesting and very well-chosen exercises. … This ‘encyclopedic’ character of the text may play an important role both as a guide to some special topics and as a source of information for both students and those whose research in related fields creates a need to familiarize themselves with the knowledge of the case when quaternion algebras are relevant.” (Juliusz Brzeziński, Mathematical Reviews, September, 2022)Table of Contents1. Introduction.- 2. Beginnings.- 3. Involutions.- 4. Quadratic Forms.- 5. Ternary Quadratic Forms.- 6. Characteristic 2.- 7. Simple Algebras.- 8. Simple Algebras and Involutions.- 9. Lattices and Integral Quadratic Forms.- 10. Orders.- 11. The Hurwitz Order.- 12. Ternary Quadratic Forms Over Local Fields.- 13. Quaternion Algebras Over Local Fields.- 14. Quaternion Algebras Over Global Fields.- 15. Discriminants.- 16. Quaternion Ideals and Invertability.- 17. Classes of Quaternion Ideals.- 18. Picard Group.- 19. Brandt Groupoids.- 20. Integral Representation Theory.- 21. Hereditary and Extremal Orders.- 22. Ternary Quadratic Forms.- 23. Quaternion Orders.- 24. Quaternion Orders: Second Meeting.- 25. The Eichler Mass Formula.- 26. Classical Zeta Functions.- 27. Adelic Framework.- 28. Strong Approximation.- 29. Idelic Zeta Functions.- 30. Optimal Embeddings.- 31. Selectivity.- 32. Unit Groups.- 33. Hyperbolic Plane.- 34. Discrete Group Actions.- 35. Classical Modular Group.- 36. Hyperbolic Space.- 37. Fundamental Domains.- 38. Quaternionic Arithmetic Groups.- 39. Volume Formula.- 40. Classical Modular Forms.- 41. Brandt Matrices.- 42. Supersingular Elliptic Curves.- 43. Abelian Surfaces with QM.

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Book SynopsisThis textbook offers an introduction to the theory of Drinfeld modules, mathematical objects that are fundamental to modern number theory.After the first two chapters conveniently recalling prerequisites from abstract algebra and non-Archimedean analysis, Chapter 3 introduces Drinfeld modules and the key notions of isogenies and torsion points. Over the next four chapters, Drinfeld modules are studied in settings of various fields of arithmetic importance, culminating in the case of global fields. Throughout, numerous number-theoretic applications are discussed, and the analogies between classical and function field arithmetic are emphasized.Drinfeld Modules guides readers from the basics to research topics in function field arithmetic, assuming only familiarity with graduate-level abstract algebra as prerequisite. With exercises of varying difficulty included in each section, the book is designed to be used as the primary textbook for a graduate course on the topic, and may also provide a supplementary reference for courses in algebraic number theory, elliptic curves, and related fields. Furthermore, researchers in algebra and number theory will appreciate it as a self-contained reference on the topic.Table of ContentsPreface.- Acknowledgements.- Notation and Conventions.- Chapter 1. Algebraic Preliminaries.- Chapter 2. Non-Archimedean Fields.- Chapter 3. Basic Properties of Drinfeld Modules.- Chapter 4. Drinfeld Modules over Finite Fields.- Chapter 5. Analytic Theory of Drinfeld Modules.- Chapter 6. Drinfeld Modules over Local Fields.- Chapter 7. Drinfeld Modules over Global Fields.- Appendix A. Drinfeld modules for general function rings.- Appendix B. Notes on exercises.- Bibliography.- Index.

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Book SynopsisThis book studies the modules arising in Fourier expansions of automorphic forms, namely Fourier term modules on SU(2,1), the smallest rank one Lie group with a non-abelian unipotent subgroup. It considers the “abelian” Fourier term modules connected to characters of the maximal unipotent subgroups of SU(2,1), and also the “non-abelian” modules, described via theta functions. A complete description of the submodule structure of all Fourier term modules is given, with a discussion of the consequences for Fourier expansions of automorphic forms, automorphic forms with exponential growth included.These results can be applied to prove a completeness result for Poincaré series in spaces of square integrable automorphic forms.Aimed at researchers and graduate students interested in automorphic forms, harmonic analysis on Lie groups, and number-theoretic topics related to Poincaré series, the book will also serve as a basic reference on spectral expansion with Fourier-Jacobi coefficients. Only a background in Lie groups and their representations is assumed.Table of Contents- 1. Introduction. - 2. The Lie Group SU(2,1) and Subgroups. - 3. Fourier Term Modules. - 4. Submodule Structure. - 5. Application to Automorphic Forms.

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Book SynopsisElliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem. The key to the conjectureTrade Review"The authors present their discussion in an informal, sometimes playful manner and with detail that will appeal to an audience with a basic understanding of calculus. This book will captivate math enthusiasts as well as readers curious about an intriguing and still unanswered question."--Margaret Dominy, Library Journal "Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics."--Mathematics Magazine "The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves."--Sungkon Chang, Times Higher Education "One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles."--James Case, SIAM News "Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection... [A]sh and Gross deliver ample and current intellectual and technical substance."--Choice "I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study."--Lisa A. Berger, Mathematical Reviews Clippings "The book is very pleasantly written, and in my opinion, the authors have done an admirable job in giving an idea to non-experts what the Birch-Swinnerton Dyer conjecture is about."--Jan-Hendrik Evertse, Zentralblatt MATH "The book's most important contributions ... are the sense of discovery, invention, and insight into the habits of mind used by mathematicians on this journey. I would recommend this book to anyone who wants to be challenged mathematically or who wants to experience mathematics as creative and exciting."--Jacqueline Coomes, Mathematics Teacher "[T]his book is a wonderful introduction to what is arguably one of the most important mathematical problems of our time and for that reason alone it deserves to be widely read. Another reason to recommend this book is the opportunity to share in the readily apparent joy the authors have for their subject and the beauty they see in it, not least because ... joy and beauty are the most important reasons for doing mathematics, irrespective of its dollar value."--Rob Ashmore, Mathematics Today "This book has many nice aspects. Ash and Gross give a truly stimulating introduction to elliptic curves and the BSD conjecture for undergraduate students. The main achievement is to make a relative easy exposition of these so technical topics."--Jonathan Sanchez-Hernandez, Mathematical SocietyTable of ContentsPreface xiii Acknowledgments xix Prologue 1 PART I. DEGREE Chapter 1. Degree of a Curve 13 1.Greek Mathematics 13 2.Degree 14 3.Parametric Equations 20 4.Our Two Definitions of Degree Clash 23 Chapter 2. Algebraic Closures 26 1.Square Roots of Minus One 26 2.Complex Arithmetic 28 3.Rings and Fields 30 4.Complex Numbers and Solving Equations 32 5.Congruences 34 6.Arithmetic Modulo a Prime 38 7.Algebraic Closure 38 Chapter 3. The Projective Plane 42 1.Points at Infinity 42 2.Projective Coordinates on a Line 46 3.Projective Coordinates on a Plane 50 4.Algebraic Curves and Points at Infinity 54 5.Homogenization of Projective Curves 56 6.Coordinate Patches 61 Chapter 4. Multiplicities and Degree 67 1.Curves as Varieties 67 2.Multiplicities 69 3.Intersection Multiplicities 72 4.Calculus for Dummies 76 Chapter 5. B'ezout's Theorem 82 1.A Sketch of the Proof 82 2.An Illuminating Example 88 PART II. ELLIPTIC CURVES AND ALGEBRA Chapter 6. Transition to Elliptic Curves 95 Chapter 7. Abelian Groups 100 1.How Big Is Infinity? 100 2.What Is an Abelian Group? 101 3.Generations 103 4.Torsion 106 5.Pulling Rank 108 Appendix: An Interesting Example of Rank and Torsion 110 Chapter 8. Nonsingular Cubic Equations 116 1.The Group Law 116 2.Transformations 119 3.The Discriminant 121 4.Algebraic Details of the Group Law 122 5.Numerical Examples 125 6.Topology 127 7.Other Important Facts about Elliptic Curves 131 5.Two Numerical Examples 133 Chapter 9. Singular Cubics 135 1.The Singular Point and the Group Law 135 2.The Coordinates of the Singular Point 136 3.Additive Reduction 137 4.Split Multiplicative Reduction 139 5.Nonsplit Multiplicative Reduction 141 6.Counting Points 145 7.Conclusion 146 Appendix A: Changing the Coordinates of the Singular Point 146 Appendix B: Additive Reduction in Detail 147 Appendix C: Split Multiplicative Reduction in Detail 149 Appendix D: Nonsplit Multiplicative Reduction in Detail 150 Chapter 10. Elliptic Curves over Q 152 1.The Basic Structure of the Group 152 2.Torsion Points 153 3.Points of Infinite Order 155 4.Examples 156 PART III. ELLIPTIC CURVES AND ANALYSIS Chapter 11. Building Functions 161 1.Generating Functions 161 2.Dirichlet Series 167 3.The Riemann Zeta-Function 169 4.Functional Equations 171 5.Euler Products 174 6.Build Your Own Zeta-Function 176 Chapter 12. Analytic Continuation 181 1.A Difference that Makes a Difference 181 2.Taylor Made 185 3.Analytic Functions 187 4.Analytic Continuation 192 5.Zeroes, Poles, and the Leading Coefficient 196 Chapter 13. L-functions 199 1.A Fertile Idea 199 2.The Hasse-Weil Zeta-Function 200 3.The L-Function of a Curve 205 4.The L-Function of an Elliptic Curve 207 5.Other L-Functions 212 Chapter 14. Surprising Properties of L-functions 215 1.Compare and Contrast 215 2.Analytic Continuation 220 3.Functional Equation 221 Chapter 15. The Conjecture of Birch and Swinnerton-Dyer 225 1.How Big Is Big? 225 2.Influences of the Rank on the Np's 228 3.How Small Is Zero? 232 4.The BSD Conjecture 236 5.Computational Evidence for BSD 238 6.The Congruent Number Problem 240 Epilogue 245 Retrospect 245 Where DoWe Go from Here? 247 Bibliography 249 Index 251

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Book SynopsisWritten by a distinguished mathematician and teacher, this undergraduate text uses a combinatorial approach to accommodate both math majors and liberal arts students. In addition to covering the basics of number theory, it offers an outstanding introduction to partitions, plus chapters on multiplicativity-divisibility, quadratic congruences, additivity, and more.

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Book SynopsisMagic tricks can be easy to perform and have an interesting mathematical foundation. In this rich, colourfully illustrated volume, Ehrhard Behrends presents around 30 card tricks and number games that are easy to learn, with no prior knowledge required. This is maths as you've never experienced it before: entertaining and fun!Table of Contents You can count on it Let's mix it up! Optimally packaged information: Coding Chance makes magic Appendix References.

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Book SynopsisJoseph Liouville is recognised as one of the great mathematicians of the nineteenth century, and one of his greatest achievements was the introduction of a powerful new method into elementary number theory. This book provides a gentle introduction to this method, explaining it in a clear and straightforward manner. The many applications provided include applications to sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums involving the divisor functions, and many others. All of the topics discussed have a rich history dating back to Euler, Jacobi, Dirichlet, Ramanujan and others, and they continue to be the subject of current mathematical research. Williams places the results in their historical and contemporary contexts, making the connection between Liouville's ideas and modern theory. This is the only book in English entirely devoted to the subject and is thus an extremely valuable resource for both students and researchers alike.Trade Review'This is the only book in English entirely devoted to the subject and is thus an extremely valuable resource for both students and researchers alike.' Mathematical ReviewsTable of ContentsPreface; 1. Joseph Liouville (1809–1888); 2. Liouville's ideas in number theory; 3. The arithmetic functions σk(n), σk*(n), dk,m(n) and Fk(n); 4. The equation i2 + jk = n; 5. An identity of Liouville; 6. A recurrence relation for σ*(n); 7. The Girard–Fermat theorem; 8. A second identity of Liouville; 9. Sums of two, four and six squares; 10. A third identity of Liouville; 11. Jacobi's four squares formula; 12. Besge's formula; 13. An identity of Huard, Ou, Spearman and Williams; 14. Four elementary arithmetic formulae; 15. Some twisted convolution sums; 16. Sums of two, four, six and eight triangular numbers; 17. Sums of integers of the form x2+xy+y2; 18. Representations by x2+y2+z2+2t2, x2+y2+2z2+2t2 and x2+2y2+2z2+2t2; 19. Sums of eight and twelve squares; 20. Concluding remarks; References; Index.

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Book SynopsisThis is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.Trade Review'This 2-volume book is a very welcome addition to the existing literature in modular representation theory. It contains a wealth of material much of which is here presented in textbook form for the first time. It gives an excellent overview of the state of the art in this fascinating subject and also of the many challenging and fundamental open problems. It is well written and will certainly become a standard reference.' Burkhard Kűlshammer, MathSciNetTable of ContentsIntroduction; 6. Blocks and source algebras; 7. Modules over finite p-groups; 8. Local structure; 9. Isometries and bimodules; 10. Structural results in block theory; 11. Blocks with cyclic defect groups; 12. Blocks with Klein four defect groups; Appendix; References; Index.

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Book SynopsisFamous mathematical constants include the ratio of circular circumference to diameter, p = 3.14 , and the natural logarithm base, e = 2.718 . Students and professionals can often name a few others, but there are many more buried in the literature and awaiting discovery. How do such constants arise, and why are they important? Here the author renews the search he began in his book Mathematical Constants, adding another 133 essays that broaden the landscape. Topics include the minimality of soap film surfaces, prime numbers, elliptic curves and modular forms, PoissonVoronoi tessellations, random triangles, Brownian motion, uncertainty inequalities, PrandtlBlasius flow (from fluid dynamics), Lyapunov exponents, knots and tangles, continued fractions, GaltonWatson trees, electrical capacitance (from potential theory), Zermelo''s navigation problem, and the optimal control of a pendulum. Unsolved problems appear virtually everywhere as well. This volume continues an outstanding scholarly atTrade Review'Like the best sequels, this one covers similar ground to the original but finds ways to stay fresh and interesting … any mathematician or math student who picks it up and spends a few minutes with it is likely to find something that is new and of interest to them. … Finch has once again written a collection of essays about a wide range of topics that I expect I will enjoy flipping through for another decade and a half until I look forward to having Volume III land on my desk.' Darren Glass, MAA reviews'This is a remarkable book … [which] can be thought of as a collection of essays that recount stories that are both successful and tangible.' Paul F. Bracken, MathSciNet'Some of the most intriguing formulas of mathematics (like those of Ramanujan) adorn this treasure trove of mathematical gems … Steven R. Finch's incredible labor of love, an encyclopedia of mathematical constants … contain a total of 269 meticulously documented essays from all fields of mathematics.' Osmo Peokonen, The Mathematical Intelligencer'Taken together, Mathematical Constants and Mathematical Constants II form a comprehensive and unique work that is a welcome addition to the mathematician's reference library.' Steven R. Finch, Notices of the AMS'Great care is taken about numerical results and the precise determination of constants. The choice of the material complements the first volume; overall, the topics seem also to be more advanced, but every now and then there is a little pearl which is indeed accessible at high school level. The text is certainly not intended for linear reading - although this might well be possible - but for eclectic readers who want to enjoy themselves and broaden their horizons, or for researchers who need information on a particular constant and further stepping stones.' Rene L. Schilling, The Mathematical Gazette'Great care is taken about numerical results and the precise determination of constants. The choice of the material complements the first volume; overall, the topics seem also to be more advanced, but every now and then there is a little pearl which is indeed accessible at high school level. The text is certainly not intended for linear reading - although this might well be possible - but for eclectic readers who want to enjoy themselves and broaden their horizons, or for researchers who need information on a particular constant and further stepping stones.' Rene L. Schilling, The Mathematical GazetteTable of Contents1. Number theory and combinatorics; 2. Inequalities and approximation; 3. Real and complex analysis; 4. Probability and stochastic processes; 5. Geometry and topology; Index.

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Book SynopsisThe Woods Hole trace formula is a Lefschetz fixed-point theorem for coherent cohomology on algebraic varieties. It leads to a version of the sheaves-functions dictionary of Deligne, relating characteristic-p-valued functions on the rational points of varieties over finite fields to coherent modules equipped with a Frobenius structure. This book begins with a short introduction to the homological theory of crystals of Böckle and Pink with the aim of introducing the sheaves-functions dictionary as quickly as possible, illustrated with elementary examples and classical applications. Subsequently, the theory and results are expanded to include infinite coefficients, L-functions, and applications to special values of Goss L-functions and zeta functions. Based on lectures given at the Morningside Center in Beijing in 2013, this book serves as both an introduction to the Woods Hole trace formula and the sheaves-functions dictionary, and to some advanced applications on characteristic p zeta vTable of ContentsIntroduction; 1. τ-sheaves, crystals, and their trace functions; 2. Functors between categories of crystals; 3. The Woods Hole trace formula; 4. Elementary applications; 5. Crystals with coefficients; 6. Cohomology of symmetric powers of curves; 7. Trace formula for L-functions; 8. Special values of L-functions; Appendix A. The trace formula for a transversal endomorphism.

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Book SynopsisThis book contains papers on topics in combinatorics (including graph theory) or number theory. The subject areas within correspond to the MSC (Mathematics Subject Classification) codes 05, 11, 20D60, and 52. Some topics discussed in this compilation include restricted Eisenstein series and certain convolution sums; zeroes of the Hurwitz zeta function in the interval (O,1); prime factorization conditions providing multiplicities in coset partitions of groups; mean value formulas for twisted Edwards curves; binary matrices as morphisms of a triangular category; some diophantine triples and quadruples for quadratic polynomials; codes associated with orthogonal groups; combinatorial sums and series involving inverses of the Gaussian binomial co-effecients; full friendly index sets and full product-cordial index sets of twisted cylinders; and properly charged coloring of two-dimensional arrays.

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Book SynopsisThis book contains papers on topics in combinatorics (including graph theory) or number theory. The subject areas within correspond to the MSC (Mathematics Subject Classification) codes 05, 11, 20D60, and 52. Some topics included in this compilation are pseudorandom binary functions on rooted plane trees; class number one criteria for real quadratic fields with discriminant k2p24p; some product-to-sum identities; a zeta function for juggling sequences; divisibility properties of hypergeometric polynomials; the distance between perfect numbers; a new proof of a theorem of Hamidoune avoiding; conjectures on the monotonicity of some arithmetical sequences; complexity of trapezoidal graphs with different triangulations; applications of shuffle products of multiple zeta values in combinatorics; the invariant area formulas and lattice point bounds for the intersection of hyperbolic and elliptic regions; and product-cordial index set for Cartesian products of a graph with a path.

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Book SynopsisThis book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski''s Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck''s duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group.The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo''s criterion is proved and also the existence of the minimal regular modeTrade ReviewWill be useful to graduate students as an introduction to arithmetic algebraic geometry, and to more advanced readers and experts in the field. * EMS *This book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive. * Zentralblatt MATH *A thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting ... The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research. * Zentralblatt MATH *Table of ContentsIntroduction ; 1. Some topics in commutative algebra ; 2. General Properties of schemes ; 3. Morphisms and base change ; 4. Some local properties ; 5. Coherent sheaves and Cech cohmology ; 6. Sheaves of differentials ; 7. Divisors and applications to curves ; 8. Birational geometry of surfaces ; 9. Regular surfaces ; 10. Reduction of algebraic curves ; Bibilography ; Index

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Book SynopsisPresents easy to understand proofs of some of the most difficult results about polynomials demonstrated by means of applications.Trade ReviewPresents easy to understand proofs of some of the most difficult results about polynomials demonstrated by means of applications ... Brings to the subject an immense range of reference to the study of polynomials. Professional and academic mathematicians of complex analysis, approximation theory and theoretical numerical analysis; graduate students in mathematics; engineers, statisticians and theoretical physicists, who have an interest in the important results about polynomials, will not do better than start with reading and referring to this book. * Current Engineering Practice *A nicely written book that will be useful for scientists, engineers and mathematicians from other fields. It can be strongly recommended as an undergraduate or graduate text and as a comprehensive source for self study. * EMS *Table of Contents2. FUNDAMENTAL RESULTS ON CRITICAL POINTS ; 8. INCLUSION OF ALL ZEROS ; 12. GROWTH ESTIMATES

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Book SynopsisFollowing a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields.Trade ReviewFrom the reviews of the second edition:"This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWS“The book under review is the second, revised, enlarged, and updated edition of J. Silverman’s meanwhile classical primer of the arithmetic of elliptic curves. … All together, this enlarged and updated version of J. Silverman’s classic ‘The Arithmetic of Elliptic Curves’ significantly increases the unchallenged value of this modern primer as a standard textbook in the field. … This makes the entire text a perfect source for teachers and students, for courses and self-study, and for further studies in the arithmetic of elliptic curves likewise.” (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010)“For the second edition of his masterly book, the author considerably updated and improved several results and proofs. … book contains a great many exercises, many of which develop or complement the results from the main body of the book. … The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. … Summarizing, this is an excellent book … . useful both for experienced mathematicians and for graduate students.”­­­ (Vasil' I. Andriĭchuk, Mathematical Reviews, Issue 2010 i)“This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves … . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory text and a standard reference. … The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)“The book is written for graduate students … and for researchers interested in standard facts about elliptic curves. … A wonderful textbook on the arithmetic theory of elliptic curves and it is a very popular introduction to the subject. … I recommend this book for anyone interested in the mathematical study of elliptic curves. It is an excellent introduction, elegant and very well written. It is one of the best textbooks to graduate level studies I have ever had contact yet.” (Book Inspections Blog, 2012)Table of ContentsAlgebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of an Elliptic Curve.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.- Computing the Mordell#x2013;Weil Group.- Algorithmic Aspects of Elliptic Curves.

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Book SynopsisMarks the 20th anniversary of the New York Number Theory Seminar (NYNTS). Beginning in 1982, the NYNTS has tried to present research in number theory and related fields of mathematics, from physics to geometry to combinatorics and computer science. The list of seminar speakers includes Fields Medallists and other established researchers.Table of ContentsThe spanning number and the independence number of a subset of an abelian group.- A formula related to the Frobenius problem in two dimensions.- One bit world.- Use of Pade approximation in spline construction.- Interactions between number theory and operator algebras in the study of Riemann zeta function (d'apres Bost-Connes and Connes).- A hyperelliptic curve with real multiplication of degree two.- Humbert's conic model and the Kummer surface.- Arithmeticity and theta correspondence of an orthogonal group.- Morphis heights and periodic points.- The elementary proof of the prime number theorem: An historical perspective.- Additive bases representations and the Erdos-Turan conjecture.- The boundary structure of the sumset in Z^2.- On NTU's in function fields.- Continued fractions and quadratic irrationals.- The inverse problem for representation functions of additive bases.- On the ubiquity of Sidon sets.

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Book SynopsisOne General Basic Theory.- I Algebraic Integers.- II Completions.- III The Different and Discriminant.- IV Cyclotomic Fields.- V Parallelotopes.- VI The Ideal Function.- VII Ideles and Adeles.- VIII Elementary Properties of the Zeta Function and L-series.- Two Class Field Theory.- IX Norm Index Computations.- X The Artin Symbol, Reciprocity Law, and Class Field Theory.- XI The Existence Theorem and Local Class Field Theory.- XII L-series Again.- Three Analytic Theory.- XIII Functional Equation of the Zeta Function, Hecke's Proof.- XIV Functional Equation, Tate's Thesis.- XV Density of Primes and Tauberian Theorem.- XVI The Brauer-Siegel Theorem.- XVII Explicit Formulas.Trade ReviewSecond Edition S. Lang Algebraic Number Theory "This book is the second edition of Lang's famous and indispensable book on algebraic number theory. The major change from the previous edition is that the last chapter on explicit formulas has been completely rewritten. In addition, a few new sections have been added to the other chapters . . . Lang's books are always of great value for the graduate student and the research mathematician. This updated edition of Algebraic number theory is no exception."—MATHEMATICAL REVIEWSTable of ContentsOne General Basic Theory.- I Algebraic Integers.- II Completions.- III The Different and Discriminant.- IV Cyclotomic Fields.- V Parallelotopes.- VI The Ideal Function.- VII Ideles and Adeles.- VIII Elementary Properties of the Zeta Function and L-series.- Two Class Field Theory.- IX Norm Index Computations.- X The Artin Symbol, Reciprocity Law, and Class Field Theory.- XI The Existence Theorem and Local Class Field Theory.- XII L-series Again.- Three Analytic Theory.- XIII Functional Equation of the Zeta Function, Hecke’s Proof.- XIV Functional Equation, Tate’s Thesis.- XV Density of Primes and Tauberian Theorem.- XVI The Brauer-Siegel Theorem.- XVII Explicit Formulas.

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Book SynopsisIn the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted."Table of Contents1.- I Elliptic and Modular Functions.- §1. The Modular Group.- §2. The Modular Curve X(1).- §3. Modular Functions.- §4. Uniformization and Fields of Moduli.- §5. Elliptic Functions Revisited.- §6. q-Expansions of Elliptic Functions.- §7. q-Expansions of Modular Functions.- §8. Jacobi’s Product Formula for ?(?).- §9. Hecke Operators.- §10. Hecke Operators Acting on Modular Forms.- §11. L-Series Attached to Modular Forms.- Exercises.- II Complex Multiplication.- §1. Complex Multiplication over C.- §2. Rationality Questions.- §3. Class Field Theory — A Brief Review.- §4. The Hilbert Class Field.- §5. The Maximal Abelian Extension.- §6. Integrality of j.- §7. Cyclotomic Class Field Theory.- §8. The Main Theorem of Complex Multiplication.- §9. The Associated Grössencharacter.- §10. The L-Series Attached to a CM Elliptic Curve.- Exercises.- III Elliptic Surfaces.- §1. Elliptic Curves over Function Fields.- §2. The Weak Mordell-Weil Theorem.- §3. Elliptic Surfaces.- §4. Heights on Elliptic Curves over Function Fields.- §5. Split Elliptic Surfaces and Sets of Bounded Height.- §6. The Mordell-Weil Theorem for Function Fields.- §7. The Geometry of Algebraic Surfaces.- §8. The Geometry of Fibered Surfaces.- §9. The Geometry of Elliptic Surfaces.- §10. Heights and Divisors on Varieties.- §11. Specialization Theorems for Elliptic Surfaces.- §12. Integral Points on Elliptic Curves over Function Fields.- Exercises.- IV The Néron Model.- §1. Group Varieties.- §2. Schemes and S-Schemes.- §3. Group Schemes.- §4. Arithmetic Surfaces.- §5. Néron Models.- §6. Existence of Néron Models.- §7. Intersection Theory, Minimal Models, and Blowing-Up.- §8. The Special Fiber of a Néron Model.- §9. Tate’s Algorithm to Compute the Special Fiber.- §10. The Conductor of an Elliptic Curve.- §11. Ogg’s Formula.- Exercises.- V Elliptic Curves over Complete Fields.- §1. Elliptic Curves over ?.- §2. Elliptic Curves over ?.- §3. The Tate Curve.- §4. The Tate Map Is Surjective.- §5. Elliptic Curves over p-adic Fields.- §6. Some Applications of p-adic Uniformization.- Exercises.- VI Local Height Functions.- §1. Existence of Local Height Functions.- §2. Local Decomposition of the Canonical Height.- §3. Archimedean Absolute Values — Explicit Formulas.- §4. Non-Archimedean Absolute Values — Explicit Formulas.- Exercises.- Appendix A Some Useful Tables.- §3. Elliptic Curves over ? with Complex Multiplication.- Notes on Exercises.- References.- List of Notation.

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Book SynopsisThe new edition of this thorough examination of the distribution of prime numbers in arithmetic progressions offers many revisions and corrections as well as a new section recounting recent works in the field.Trade ReviewFrom the reviews of the third edition: "The book under review is one of the most important references in the multiplicative number theory, as its title mentions exactly. … Davenport’s book covers most of the important topics in the theory of distribution of primes and leads the reader to serious research topics … . is very well written. … is useful for graduate students, researchers and for professors. It is a very good text source specially for graduate levels, but even is fruitful for undergraduates." (Mehdi Hassani, MathDL, July, 2008)Table of ContentsFrom the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The Distribution of the Primes.- Riemann's Memoir.- The Functional Equation of the L Function.- Properties of the Gamma Function.- Integral Functions of Order 1.- The Infinite Products for xi(s) and xi(s,Zero-Free Region for zeta(s).- Zero-Free Regions for L(s, chi).- The Number N(T).- The Number N(T, chi).- The explicit Formula for psi(x).- The Prime Number Theorem.- The Explicit Formula for psi(x,chi).- The Prime Number Theorem for Arithmetic Progressions (I).- Siegel's Theorem.- The Prime Number Theorem for Arithmetic Progressions (II).- The Pólya-Vinogradov Inequality.- Further Prime Number Sums.

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Book Synopsis1 Elementary Concepts.- 2 Reduction of Positive Definite Forms.- 3 Indefinite Forms.- 3.1 Reduction, Cycles.- 3.2 Automorphs, Pell's Equation.- 3.3 Continued Fractions and Indefinite Forms.- 4 The Class Group.- 4.1 Representation and Genera.- 4.2 Composition Algorithms.- 4.3 Generic Characters Revisited.- 4.4 Representation of Integers.- 5 Miscellaneous Facts.- 5.1 Class Number Computations.- 5.2 Extreme Cases and Asymptotic Results.- 6 Quadratic Number Fields.- 6.1 Basic Algebraic Definitions.- 6.2 Algebraic Numbers and Quadratic Fields.- 6.3 Ideals in Quadratic Fields.- 6.4 Binary Quadratic Forms and Classes of Ideals.- 6.5 History.- 7 Composition of Forms.- 7.1 Nonfundamental Discriminants.- 7.2 The General Problem of Composition.- 7.3 Composition in Different Orders.- 8 Miscellaneous Facts II.- 8.1 The Cohen-Lenstra Heuristics.- 8.2 Decomposing Class Groups.- 8.3 Specifying Subgroups of Class Groups.- 9 The 2-Sylow Subgroup.- 9.1 Classical Results on the Pell Equation.- 9.2 ModernRTable of Contents1 Elementary Concepts.- 2 Reduction of Positive Definite Forms.- 3 Indefinite Forms.- 3.1 Reduction, Cycles.- 3.2 Automorphs, Pell’s Equation.- 3.3 Continued Fractions and Indefinite Forms.- 4 The Class Group.- 4.1 Representation and Genera.- 4.2 Composition Algorithms.- 4.3 Generic Characters Revisited.- 4.4 Representation of Integers.- 5 Miscellaneous Facts.- 5.1 Class Number Computations.- 5.2 Extreme Cases and Asymptotic Results.- 6 Quadratic Number Fields.- 6.1 Basic Algebraic Definitions.- 6.2 Algebraic Numbers and Quadratic Fields.- 6.3 Ideals in Quadratic Fields.- 6.4 Binary Quadratic Forms and Classes of Ideals.- 6.5 History.- 7 Composition of Forms.- 7.1 Nonfundamental Discriminants.- 7.2 The General Problem of Composition.- 7.3 Composition in Different Orders.- 8 Miscellaneous Facts II.- 8.1 The Cohen-Lenstra Heuristics.- 8.2 Decomposing Class Groups.- 8.3 Specifying Subgroups of Class Groups.- 8.3.1 Congruence Conditions.- 8.3.2 Exact and Exotic Groups.- 9 The 2-Sylow Subgroup.- 9.1 Classical Results on the Pell Equation.- 9.2 Modern Results.- 9.3 Reciprocity Laws.- 9.4 Special References for Chapter 9.- 10 Factoring with Binary Quadratic Forms.- 10.1 Classical Methods.- 10.2 SQUFOF.- 10.3 CLASNO.- 10.4 SPAR.- 10.4.1 Pollard p — 1.- 10.4.2 SPAR.- 10.5 CFRAC.- 10.6 A General Analysis.- Appendix 1:Tables, Negative Discriminants.- Appendix 2:Tables, Positive Discriminants.

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Book SynopsisSome of the central topics in number theory, presnted in a simple and concise fashion. All the topics are presented in a refreshingly elegant and efficient manner with clever examples and interesting problems throughout. The text is suitable for a graduate course in analytic number theory.Trade ReviewFrom the reviews:D. J. NewmanAnalytic Number Theory"This book is remarkable . . . The author’s style remains pleasantly discursive throughout the book. Any of these chapters might be useful to a reader planning a lecture course in the relevant subject area . . . The student of analytic number theory would do well to find shelf-room for this book."—MATHEMATICAL “Donald J. Newman was a noted problem-solver who believed that math should be fun and that beautiful theorems should have beautiful proofs. This short book collects brief, self-contained proofs of several well-known theorems in analytic number theory … .” (Allen Stenger, The Mathematical Association of America, November, 2010)Table of ContentsThe Idea of Analytic Number Theory.- The Partition Function.- The Erd?s-Fuchs Theorem.- Sequences without Arithmetic Progressions.- The Waring Problem.- A “Natural” Proof of the Nonvanishing of L-Series.- Simple Analytic Proof of the Prime Number Theorem.

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Book SynopsisIt offers many features rarely treated in introductory p-adic texts such as topological models of p-adic spaces inside Euclidian space, a special case of Hazewinkel’s functional equation lemma, and a treatment of analytic elements.Trade Review“This well-written book, complete with all proofs and a wealth of exercises, is perfectly suited as a text book for introductory courses.” (Wim Schikhof, zbMATH 0947.11035, 2022)From the reviews: MATHEMATICAL REVIEWS "The text ends with a large number of exercises. The writing is extremely clear and very meticulous. The bibliography, which does not attempt to be comprehensive, is adequate. I recommend A. Robert’s book without reservation to anyone who wants to have a reference text on one-variable p-adic analysis that is clear, complete and pleasant to read." MATHSCINET "Robert's book is aimed at an intermediate level between the very specialized monographs and the elementary texts. It has no equal in the marketplace, because it covers practically all of p-adic analysis of one variable (except the rationality of the zeta function of an algebraic variety over a finite field and the theory of p-adic differential equations) and contains numerous results that were accessible only in articles or even in preprints. ...I recommend A. Robert's book without reservation to anyone who wants to have a reference text on one-variable p-adic analysis that is clear, complete and pleasant to read."D. Barsky in MathSciNet, August 2001Table of Contents1 p-adic Numbers.- 2 Finite Extensions of the Field of p-adic Numbers.- 3 Construction of Universal p-adic Fields.- 4 Continuous Functions on Zp.- 5 Differentiation.- 6 Analytic Functions and Elements.- 7 Special Functions, Congruences.- Specific References for the Text.- Tables.- Basic Principles of Ultrametric Analysis.- Conventions, Notation, Terminology.

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