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# Arithmetic Differential Equations

Alexandru Buium

310 pages, parution le 27/10/2005

## Résumé

This monograph contains exciting original mathematics that will inspire new directions of research in algebraic geometry. Developed here is an arithmetic analog of the theory of ordinary differential equations, where functions are replaced by integer numbers, the derivative operator is replaced by a "Fermat quotient operator", and differential equations (viewed as functions on jet spaces) are replaced by "arithmetic differential equations". The main application of this theory concerns the construction and study of quotients of algebraic curves by correspondences with infinite orbits. Any such quotient reduces to a point in algebraic geometry. But many of the above quotients cease to be trivial (and become quite interesting) if one enlarges algebraic geometry by using arithmetic differential equations in place of algebraic equations.

This book, in part, follows a series of papers written by the author. However, a substantial amount of the material has never been published before. For most of the book, the only prerequisites are the basic facts of algebraic geometry and algebraic number theory. It is suitable for graduate students and researchers interested in algebraic geometry and number theory.

Graduate students and research mathematicians interested in algebraic geometry and number theory.

## Sommaire

• Main concepts and results
• Preliminaries from algebraic geometry
• Outline of $\delta$-geometry
• General theory
• Global theory
• Local theory
• Birational theory
• Applications
• Spherical correspondences
• Flat correspondences
• Hyperbolic correspondences
• List of results
• Bibliography
• Index
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## Caractéristiques techniques

 PAPIER Éditeur(s) American Mathematical Society (AMS) Auteur(s) Alexandru Buium Collection Mathematical Surveys and Monographs Parution 27/10/2005 Nb. de pages 310 Format 18 x 26 Couverture Relié Poids 824g Intérieur Noir et Blanc EAN13 9780821838624 ISBN13 978-0-8218-3862-4

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