A Course in p adic Analysis

Author: Alain M. Robert
Publisher: Springer Science & Business Media
ISBN: 1475732546
Format: PDF, ePub, Mobi
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Discovered at the turn of the 20th century, p-adic numbers are frequently used by mathematicians and physicists. This text is a self-contained presentation of basic p-adic analysis with a focus on analytic topics. It 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.

A Course in p adic Analysis

Author: Alain Robert
Publisher: Springer Science & Business Media
ISBN: 9780387986692
Format: PDF, ePub, Docs
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Kurt Hensel (1861-1941) discovered the p-adic numbers around the turn of the century. These exotic numbers (or so they appeared at first) are now well-established in the mathematical world and used more and more by physicists as well. This book offers a self-contained presentation of basic p-adic analysis. The author is especially interested in the analytical topics in this field. Some of the features which are not treated in other introductory p-adic analysis texts are topological models of p-adic spaces inside Euclidean space, a construction of spherically complete fields, a p-adic mean value theorem and some consequences, a special case of Hazewinkel's functional equation lemma, a remainder formula for the Mahler expansion, and most importantly a treatment of analytic elements.

p adic Numbers p adic Analysis and Zeta Functions

Author: Neal Koblitz
Publisher: Springer Science & Business Media
ISBN: 1461211123
Format: PDF, ePub, Docs
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The first edition of this work has become the standard introduction to the theory of p-adic numbers at both the advanced undergraduate and beginning graduate level. This second edition includes a deeper treatment of p-adic functions in Ch. 4 to include the Iwasawa logarithm and the p-adic gamma-function, the rearrangement and addition of some exercises, the inclusion of an extensive appendix of answers and hints to the exercises, as well as numerous clarifications.

P adic Analysis Compared with Real

Author: Svetlana Katok
Publisher: American Mathematical Soc.
ISBN: 082184220X
Format: PDF, Mobi
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The book gives an introduction to $p$-adic numbers from the point of view of number theory, topology, and analysis. Compared to other books on the subject, its novelty is both a particularly balanced approach to these three points of view and an emphasis on topics accessible to undergraduates. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity of isometries of compact metric spaces) are also included in the book. They will enhance the reader's understanding of real analysis and intertwine the real and $p$-adic contexts of the book. The book is based on an advanced undergraduate course given by the author. The choice of the topic was motivated by the internal beauty of the subject of $p$-adic analysis, an unusual one in the undergraduate curriculum, and abundant opportunities to compare it with its much more familiar real counterpart. The book includes a large number of exercises. Answers, hints, and solutions for most of them appear at the end of the book. Well written, with obvious care for the reader, the book can be successfully used in a topic course or for self-study.

Ultrametric Calculus

Author: W. H. Schikhof
Publisher: Cambridge University Press
ISBN: 0521032873
Format: PDF, Kindle
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This is an introduction to p-adic analysis which is elementary yet complete and which displays the variety of applications of the subject. Dr Schikhof is able to point out and explain how p-adic and 'real' analysis differ. This approach guarantees the reader quickly becomes acquainted with this equally 'real' analysis and appreciates its relevance. The reader's understanding is enhanced and deepened by the large number of exercises included throughout; these both test the reader's grasp and extend the text in interesting directions. As a consequence, this book will become a standard reference for professionals (especially in p-adic analysis, number theory and algebraic geometry) and will be welcomed as a textbook for advanced students of mathematics familiar with algebra and analysis.

P adic Analysis

Author: Neal Koblitz
Publisher: Cambridge University Press
ISBN: 9780521280600
Format: PDF, ePub
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An introduction to recent work in the theory of numbers and its interrelation with algebraic geometry and analysis.

A Course in Arithmetic

Author: J-P. Serre
Publisher: Springer Science & Business Media
ISBN: 1468498843
Format: PDF, ePub, Mobi
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This book is divided into two parts. The first one is purely algebraic. Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem). It is achieved in Chapter IV. The first three chapters contain some preliminaries: quadratic reciprocity law, p-adic fields, Hilbert symbols. Chapter V applies the preceding results to integral quadratic forms of discriminant ± I. These forms occur in various questions: modular functions, differential topology, finite groups. The second part (Chapters VI and VII) uses "analytic" methods (holomor phic functions). Chapter VI gives the proof of the "theorem on arithmetic progressions" due to Dirichlet; this theorem is used at a critical point in the first part (Chapter Ill, no. 2.2). Chapter VII deals with modular forms, and in particular, with theta functions. Some of the quadratic forms of Chapter V reappear here. The two parts correspond to lectures given in 1962 and 1964 to second year students at the Ecole Normale Superieure. A redaction of these lectures in the form of duplicated notes, was made by J.-J. Sansuc (Chapters I-IV) and J.-P. Ramis and G. Ruget (Chapters VI-VII). They were very useful to me; I extend here my gratitude to their authors.

Introduction to Cyclotomic Fields

Author: Lawrence C. Washington
Publisher: Springer Science & Business Media
ISBN: 1461219345
Format: PDF, Kindle
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This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat’s Last Theorem, and Iwasawa’s theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott’s proof of the vanishing of Iwasawa’s f-invariant.

Diophantine Geometry

Author: Marc Hindry
Publisher: Springer Science & Business Media
ISBN: 1461212103
Format: PDF
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This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

p adic Numbers

Author: Fernando Q. Gouvea
Publisher: Springer Science & Business Media
ISBN: 3662222787
Format: PDF, ePub, Mobi
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p-adic numbers are of great theoretical importance in number theory, since they allow the use of the language of analysis to study problems relating toprime numbers and diophantine equations. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. The book should be of use to students interested in number theory, but at the same time offers an interesting example of the many connections between different parts of mathematics. The book strives to be understandable to an undergraduate audience. Very little background has been assumed, and the presentation is leisurely. There are many problems, which should help readers who are working on their own (a large appendix with hints on the problem is included). Most of all, the book should offer undergraduates exposure to some interesting mathematics which is off the beaten track. Those who will later specialize in number theory, algebraic geometry, and related subjects will benefit more directly, but all mathematics students can enjoy the book.