Number Theory: Volume II: Analytic and Modern Tools (Graduate Texts in Mathematics, 240) 2007th Edition by Henri Cohen | (PDF) Free Download

Description

This book deals with several aspects of what is now called “explicit number theory.” The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.

User’s Reviews

Editorial Reviews: Review From the reviews:”Cohen (Université Bordeaux I, France), an instant classic, uniquely bridges the gap between old-fashioned, naive treatments and the many modern books available that develop the tools just mentioned … . Summing Up: Recommended. … Upper-division undergraduates through faculty.” (D. V. Feldman, CHOICE, Vol. 45 (5), January, 2008)”The book deals with aspects of ‘explicit number theory’. … The central theme … is the solution of Diophantine equations. … It combines an interesting ‘philosophy’ of the subject with an encyclopedic grasp of detail. The extension of the author’s reach via the contributed chapters is a good idea. Perhaps it is the start of a trend, as the subject grows more and more. … It will undoubtedly be mined by instructors for their graduate courses, particularly for the purpose of including some recently-proved content.” (R. C. Baker, Mathematical Reviews, Issue 2008 e)“This is the second volume of a highly impressive two-volume textbook on Diophantine analysis. … readers are presented with an almost overwhelming amount of material. This … text book is bound to become an important reference for students and researchers alike.” (C. Baxa, Monatshefte für Mathematik, Vol. 157 (2), June, 2009) From the Back Cover The central theme of this graduate-level number theory textbook is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three aspects. The first is the local aspect: one can do analysis in p-adic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second is the global aspect: the use of number fields, and in particular of class groups and unit groups. This classical subject is here illustrated through a wide range of examples. The third aspect deals with specific classes of equations, and in particular the general and Diophantine study of elliptic curves, including 2 and 3-descent and the Heegner point method. These subjects form the first two parts, forming Volume I. The study of Bernoulli numbers, the gamma function, and zeta and L-functions, and of p-adic analogues is treated at length in the third part of the book, including many interesting and original applications. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five chapters on these techniques forming the fourth part, which together with the third part forms Volume II. These chapters were written by Yann Bugeaud, Guillaume Hanrot, Maurice Mignotte, Sylvain Duquesne, Samir Siksek, and the author, and contain material on the use of Galois representations, points on higher-genus curves, the superfermat equation, Mihailescu’s proof of Catalan’s Conjecture, and applications of linear forms in logarithms. The book contains 530 exercises of varying difficulty from immediate consequences of the main text to research problems, and contain many important additional results.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Although the main theme of this two volume treatise is the solution of Diophantine equations, the first two chapters in the second volume (chapters 9 and 10 of the treatise) have well presented material in analytic number theory that appears in few other works. Chapter 9 is on the Bernoulli polynomials and Bernoulli numbers, the Euler-Maclaurin summation formula, the gamma function, the Mellin transform, and Bessel functions. You would probably be surprised to find how few books actually prove the Euler-Maclaurin summation formula, or work out properties of Bessel functions from scratch rather than merely deferring to Watson’s overwhelming tome (or citing some table of identities and approximations which does not actually prove the results it states). Chapter 10 is on analytic properties of Dirichlet series and L-functions. The proof of the Voronoi summation formula is clean and the best that I have found anywhere. Other topics in Chapter 10 that are not well presented in many other works are real analytic Eisenstein series, the Kronecker limit formula, and the Chowla-Selberg formula. The only other book in which I have seen the Chowla-Selberg formula proved is Weil’s “Elliptic Functions according to Eisenstein and Kronecker”, which is much harder to understand than Cohen’s book.

⭐A very good book very well-written in number theory. I recommend it for the beginners and for the specialists in this branch of mathematics.

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Free Download Number Theory: Volume II: Analytic and Modern Tools (Graduate Texts in Mathematics, 240) 2007th Edition in PDF format
Number Theory: Volume II: Analytic and Modern Tools (Graduate Texts in Mathematics, 240) 2007th Edition PDF Free Download
Download Number Theory: Volume II: Analytic and Modern Tools (Graduate Texts in Mathematics, 240) 2007th Edition 2007 PDF Free
Number Theory: Volume II: Analytic and Modern Tools (Graduate Texts in Mathematics, 240) 2007th Edition 2007 PDF Free Download
Download Number Theory: Volume II: Analytic and Modern Tools (Graduate Texts in Mathematics, 240) 2007th Edition PDF
Free Download Ebook Number Theory: Volume II: Analytic and Modern Tools (Graduate Texts in Mathematics, 240) 2007th Edition