Foundations of Arithmetic Differential Geometry
About this Title
Alexandru Buium, University of New Mexico, Albuquerque, NM
Publication: Mathematical Surveys and Monographs
Publication Year: 2017; Volume 222
ISBNs: 978-1-4704-3623-0 (print); 978-1-4704-4089-3 (online)
MathSciNet review: MR3643159
MSC: Primary 14G20; Secondary 11E57, 12H05, 53-02, 53C05, 58Axx
The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices.
One of the main conclusions of the theory is that the spectrum of the integers is “intrinsically curved”; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.
Graduate students and researchers interested in algebraic geometry, number theory, and algebraic groups.
Table of Contents
- Algebraic background
- Classical differential geometry revisited
- Arithmetic differential geometry: Generalities
- Arithmetic differential geometry: The case of $GL_n$
- Curvature and Galois groups of Ehresmann connections
- Curvature of Chern connections
- Curvature of Levi-Cività connections
- Curvature of Lax connections
- Open problems