The central theme of this graduatelevel 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 padic 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 3descent
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 Lfunctions,
and of padic 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 highergenus 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)
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