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 we start 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, I have included five chapters on these techniques forming the fourth part, which together with the third part forms Volume II. These appendices were written by Yann Bugeaud, Guillaume Hanrot, Maurice Mignotte, Sylvain Duquesne, Samir Siksek, and myself, 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.
Brief Table of Contents (Chapter Titles)
Click here for the preface and table of contents
Click here for the errataPlease send comments and errata to
Copied with permission from Joseph Silverman's html page