Number Theory Volume II Analytic and Modern Tools [Paperback]

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.

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.

Analytic Tools.- Bernoulli Polynomials and the Gamma Function.- Dirichlet Series and L-Functions.- p-adic Gamma and L-Functions.- Modern Tools.- Applications of Linear Forms in Logarithms.- Rational Points on Higher-Genus Curves.- The Super-Fermat Equation.- The Modular Approach to Diophantine Equations.- Catalans Equation.

From the reviews:

Cohen (Universit? 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 authors reach via the contributed chapters is a good idea. Perhaps it is the start of a trend, as the subl#«