Arithmetic Differential Equations (mathematical Surveys And Monographs)
by Alexandru Buium /
2005 / English / PDF
24.8 MB Download
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 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.