Arithmetic Differential Equations
Alexandru Buium
Jan. 2005 · Mathematical Surveys and Monographs Boek 118 · American Mathematical Soc.
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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 correspondence with infinite orbits. Any such quotient usually 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.
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