معادلات دیفرانسیل حسابی (پژوهش‌ها و مونوگراف‌های ریاضی، جلد ۱۱۸)

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. This Research Monograph Develops An Arithmetic Analogue Of The Theory Of Ordinary Differential Equations: Functions Are Replaced Here 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 Usual Algebraic Geometry. But Many Quotients As Above Cease To Be Trivial (and Become Quite Interesting) If One Enlarges Algebraic Geometry By Using Arithmetic Differential Equations In Place Of Algebraic Equations. The Book Partly Follows A Series Of Papers Written By The Author; However, A Substantial Part Of The Material Presented Here Has Never Been Published Before. For Most Of The Book The Only Prerequisites Are The Basic Facts Of Algebraic Geometry And Number Theory.--book Jacket. Pt. 1. Main Concepts And Results -- Ch. 1. Preliminaries From Algebraic Geometry -- Ch. 2. Outline Of [delta] -- Geometry -- Pt. 2. General Theory -- Ch. 3. Global Theory -- Ch. 4. Local Theory -- Ch. 5. Birational Theory -- Pt. 3. Applications -- Ch. 6. Spherical Correspondences -- Ch. 7. Flat Correspondences -- Ch. 8. Hyperbolic Correspondences. Alexandru Buium. Includes Bibliographical References (p. 301-305) And Index. Cover Page......Page 1 Title Page......Page 2 ISBN 0821838628......Page 3 Introduction......Page 5 Index......Page 6 Part 1. Main concepts and results......Page 32 1. Preliminaries from algebraic geometry......Page 34 2. Outline of δ-geometry......Page 62 Part 2. General theory......Page 100 3. Global theory......Page 102 4. Local theory......Page 138 5. Birational theory......Page 172 Part 3. Applications......Page 190 6. Spherical correspondences......Page 192 7. Flat correspondences......Page 216 8. Hyperbolic correspondences......Page 258 List of Results......Page 330 Bibliography......Page 332 Index of terminology......Page 338 Index of notation......Page 339