Cohen Number Theory

Volume I: Tools and Diophantine Equations
1. Auflage 2008
ISBN: 978-0-387-49923-9
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 239, 673 Seiten, eBook

Reihe: Graduate Texts in Mathematics

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 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 aspect is the global aspect: the use of number fields, and in particular of class groups and unit groups. 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, and embodies in a beautiful way the local and global aspects of Diophantine problems. In fact, these functions are defined through the local aspects of the problems, but their analytic behavior is intimately linked to the global aspects. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included 5 appendices on these techniques. These appendices were written by Henri Cohen, Yann Bugeaud, Maurice Mignotte, Sylvain Duquesne, and Samir Siksek, and contain material on the use of Galois representations, the superfermat equation, Mihailescu’s proof of Catalan’s Conjecture, and applications of linear forms in logarithms.

Zielgruppe

Research

Autoren/Hrsg.

Cohen, Henri

Inhaltsverzeichnis

to Diophantine Equations.- to Diophantine Equations.- Tools.- Abelian Groups, Lattices, and Finite Fields.- Basic Algebraic Number Theory.- p-adic Fields.- Quadratic Forms and Local-Global Principles.- Diophantine Equations.- Some Diophantine Equations.- Elliptic Curves.- Diophantine Aspects of Elliptic Curves.