## 2: Number Theory: Volume II: Analytic and Modern Tools (Graduate Texts in Mathematics)

### Henri Cohen

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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.

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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.

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)

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## 1.Number Theory. Vol. 2 Analytic and Modern Tools

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## 2.Number Theory: Volume II: Analytic and Modern Tools (Hardback)

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Book Description Springer-Verlag New York Inc., United States, 2007. Hardback. Book Condition: New. Language: English . Brand New Book. 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. Bookseller Inventory # AAZ9780387498935

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## 3.Number Theory: Volume II: Analytic and Modern Tools (Hardback)

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Book Description Springer-Verlag New York Inc., United States, 2007. Hardback. Book Condition: New. Language: English . Brand New Book. 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. Bookseller Inventory # AAZ9780387498935

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## 4.NUMBER THEORY

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## 5.Number Theory

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Book Description Springer-Verlag New York Inc., 2007. Book Condition: New. This book deals with several aspects of what is now called "explicit number theory." The central theme is the solution of Diophantine equations. This unique collection of topics contains more than 350 exercises and its text is largely self-contained. Series: Graduate Texts in Mathematics. Num Pages: 596 pages, biography. BIC Classification: PBH. Category: (UU) Undergraduate. Dimension: 168 x 245 x 40. Weight in Grams: 1082. . 2007. 2007th Edition. Hardcover. . . . . . Bookseller Inventory # V9780387498935

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## 6.Number Theory

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Book Description Springer-Verlag New York Inc. Book Condition: New. This book deals with several aspects of what is now called "explicit number theory." The central theme is the solution of Diophantine equations. This unique collection of topics contains more than 350 exercises and its text is largely self-contained. Series: Graduate Texts in Mathematics. Num Pages: 596 pages, biography. BIC Classification: PBH. Category: (UU) Undergraduate. Dimension: 168 x 245 x 40. Weight in Grams: 1082. . 2007. 2007th Edition. Hardcover. . . . . Books ship from the US and Ireland. Bookseller Inventory # V9780387498935

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## 7.Number Theory: Volume II: Analytic and Modern Tools: Analytic and Modern Tools v. 2 (Graduate Texts in Mathematics)

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## 8.2: Number Theory: Volume II: Analytic and Modern Tools: Analytic and Modern Tools v. 2 (Graduate Texts in Mathematics)

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## 9.2: Number Theory: Volume II: Analytic and Modern Tools: Analytic and Modern Tools v. 2 (Graduate Texts in Mathematics)

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## 10.Number Theory : Volume II: Analytic and Modern Tools

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Book Description Springer, 2016. Paperback. Book Condition: New. PRINT ON DEMAND Book; New; Publication Year 2016; Not Signed; Fast Shipping from the UK. No. book. Bookseller Inventory # ria9780387498935_lsuk