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Foundations of Arithmetic Differential Geometry

Alexandru Buium University of New Mexico, Albuquerque, NM
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Hardcover ISBN: 978-1-4704-3623-0
Product Code: SURV/222
List Price: $124.00 MAA Member Price:$111.60
AMS Member Price: $99.20 Electronic ISBN: 978-1-4704-4089-3 Product Code: SURV/222.E List Price:$124.00
MAA Member Price: $111.60 AMS Member Price:$99.20
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List Price: $186.00 MAA Member Price:$167.40
AMS Member Price: $148.80 Click above image for expanded view Foundations of Arithmetic Differential Geometry Alexandru Buium University of New Mexico, Albuquerque, NM Available Formats:  Hardcover ISBN: 978-1-4704-3623-0 Product Code: SURV/222  List Price:$124.00 MAA Member Price: $111.60 AMS Member Price:$99.20
 Electronic ISBN: 978-1-4704-4089-3 Product Code: SURV/222.E
 List Price: $124.00 MAA Member Price:$111.60 AMS Member Price: $99.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$186.00 MAA Member Price: $167.40 AMS Member Price:$148.80
• Book Details

Mathematical Surveys and Monographs
Volume: 2222017; 344 pp
MSC: Primary 11; 14; 53;

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.

• Chapters
• Introduction
• 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

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2222017; 344 pp
MSC: Primary 11; 14; 53;

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.

• Chapters
• Introduction
• 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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